The dependence of the charge exchange lifetimes on the mirror latitude for ions mirroring off the geomagnetic equator was re-computed using the improved hydrogen distribution models. The Chamberlain model was used to define the spatial distribution of the neutral hydrogen environment through which the ring current ions traverse. The resultant dependence of the charge exchange lifetime on mirror latitude is best fitted by the approximation that contains the charge exchange lifetime for equatorial particles

Bewtra, N. K.

Smith, P. H.

English

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i	 i
X-626-76-115
DEPENDENCE OF THE CHARGE
EXCHANGE LIFETIMES ON
MIRROR LATITUDE
	
PAUL H. SMITH	 -`
	
N. K. BEWTRA	 }
(NASA-Ti9-X-71126) OLPEND6rXE OF THE CiiAPrF	 N76-26135
EXCHANGE LIFLT[MES ON MIRE?HR LATITUDE. MASAI
21 P fir $ 3.50	 CSCL 03f1	 UNCL AS
63/93 21172
JUNE 1976
OPIUM SPACE FLIGHT CENTER --°
GREENBELT, MARYLAND
5•	 `	 1
	nA. ova a
	
c
Dependence of the Charge Exchange Lifetimes
on Mirror Latitude
{
r
Paul H. Smith
Laboratory for Planetary Atmospheres
NASA/Goddard Space Flight Center
Greenbelt, Maryland 20771
N. K. Bewtra
Computer Sciences Corporation
Silver Spring, Maryland 20910
ABSTRACT
The dependence of the charge exchange lifetimes on the mirror
latitude for ions mirroring off the geomagnetic equator has been
re-computed using the improved hydrogen distribution models which have
become available since the earlier calculation by Liemohn. The 	 a
Chamberlain model, with the input parameters determined by recent satellite
observations, has been used to define the spatial distribution of the
=x
neutral hydrogen environment through which the ring current ions traverse.
3
The resultant dependence of the charge exchange lifetime, T, on mirror	 [is
latitude, 7,m , is best fit by the approximation Tm = Te cos3.5am , where
T  is the charge exchange lifetime for the equatorial particles. This
is a significant change from the cos 5am approximation of the previous 	 a
results.
f	 ,	 ,
The relative importance of the charge exchange mechanism in the
recovery of the geomagnetic storm time ring current process has been
considered in numerous work since the early suggestion by Dessler and
Parker (1959) that this mechanism could be significant. The lifetimes
calculated by Liemohn (1961) for the tens of keV ring current protons
had been the primary theoretical source for comparing observational
results with the charge exchange mechanism (Frank, 1967; Swisher and
Frank, 1968; Russell and Thorne, 1970). As Smith et al. (1976) have
pointed out, there now exists both more precise cross sections for
charge exchange of protons and atomic hydrogen (McClure, 1966), and a
much improved hydrogen distribution model (8pik and Singer, 1961;
Chamberlain, 1963), which has been used by Prolss (1973) in computing
the proton lifetimes. The dependence of the charge exchange lifetimes
on the mirror latitude for ions mirroring off the geomagnetic equator
has not been re-examined in light of the improved hydrogen distribution
models,and even the most recent articles (Tinsley, 1976; Lyons and Evans,
1976) in the literature still employ the approximation given by Liemohn
(1961) under the assumption of the neutral hydrogen density model of
Johnson and Fish (1960).
In this letter we use the Chamberlain (1963) model Opik and Singer,
1961) to provide the spatial distribution of the neutral hydrogen
environment through which the ions traverse. This model appears to be
in good agreement with recent Apollo 16 Lyman alpha imagery of the
hydrogen geocorona (Carruthers, 1976) and Lyman alpha airglow measure-
ments aboard several satellites (Meier and Mange,1970; Bertaux.and Blamont,
1973; Meier and Mange, 1973). The model assumes a spherically symmetric
j
hydrogen density and is characterized by three parameters: the tempera-
ture and density at the exobase, and the satellite critical altitude.
Critical satellite altitude, R sc , is defined to be that altitude below
which there exist closed orbit satellite particles, and above which the
creation of satellite orbits is negligible so that the only orbits pre-
sent are those that have perigees between the exobase and the critical
satellite altitude. In order to determine the functional relationship of
the charge exchange lifetime of ions mirroring off the geomagnetic
equator, Tm , to the lifetime of equatoriaily mirroring ions, T
e
o we
performed a full numerical integration for the mirroring particle motion,
and fitted the computed ratio, Tm/Te , to a function of the form cos t m,
where am is the mirror latitude. The dipole geomagnetic field is assumed
in our calculations.
The mean lifetime, Te , for protons or any other species of ions con-
fined to the equatorial plane for charge exchange decay with atomic hydro-
gen is given.by
Te n ro av
where n(ro ) is the hydrogen density in the equatorial plane, v is the
ion velocity and a is the charge exchange cross section for that ion
species. However, an ion oscillating between its mirror points, M and
M', Figure 1, encounters a variable instantaneous density, n(r), of
neutral hydrogen atoms ( Chamberlain, 1963) and, alsn, the time it needs
9
to traverse an element of length along its guiding field line varies with
the changing magnetic latitude. Since the ions make a large number of
oscillations before charge exchanging with the neutral hydrogen, we can
effectively average the instantaneous hydrogen density over a mirror path
to get the time averaged density, n, that the particle encounters. This
j	 is expressed as
-2-
i
l
/mn(r)  1(—) ds
n 
r	
(2)n =	 ^
r m	
v
1
J	 vii (r	 ds
X=o
The above integration is performed along the magnetic field line
r = ro cos ta
	
(3)
and so,	 ds = ro cosa 4-3cos 2a da	 (4)
The velocity parallel to the field line, v ii (r), is expressed
for the dipole field ( Roederer, 1970) as:
—
2
a) sin g 
ao	 (5)
costa
where ao is the equatorial pitch angle and is related to the mirror
latitude as
2	 cos6am
sin a ° =
	
	 (6)
 ^4-3co
The mean lifetime, Tm , of the mirroring particles then is given by
v^^ (r) = v
-3-
Ilf
C	
C	 ^
1
Tm = n av	 (7)
We note that the denominator of equation (2) is one quarter of
the bounce period and its evaluation provides us with a consistency
check on the numerical integration techniques to be discussed later.
The denominator integral to be evaluated thus is simply
Am
f(a ) = I cosA (4-3cos 2A)h dAo	 L1 _ sing a_n (4-3cos2A)h ]'^ 	 (8)
A=o L
	 cos6A
This integral was first evaluated by Hamlin (1961) and it
has a singularity when A=Am. The present calculation includes
derivation of this integral only as a check on the evaluation of the
numerator in equation (2). If we introduce a new variable propor-
tional to the magnetic field
X = 4- 3co s2A sing ao	 (9)
Cos6A
equation (8) reduces to a form which can be implicitly represented as
1
f(ao ) =	 -!(LL dx
f 1-x
sin g ao
-4-
i
I
I
^	 I
(10)
Krylov (1962) has developed very powerful techniques for numerical
integration using Orthogonal Polynomials. It can be shown (Abramowitz
and Stegun, 1965) that
fb 
21Ady =	 b-a	 wi g (y i) + Rn	 (11)
a 
% b-y	 i=1
where
yi = a + (b-a)xi
x i = 1 - ^ j
gi = ith positive root of Legendre polynomial P2n(x)
wi = 2 wi(2n), wi(2n) are the Gaussian weights of order 2n
__ 24n+1	 [(2n)133	 (2n)
Rn 4n+1
	 [(`4n)^ 9	
(n), a < n < b .
Thus, the integral is evaluated exactly whenever g(y) is a polynomial
of degree (2n-1) or less. In our integration we use n = 32. In Table I
we show the excellent agreement between the function f(ao ) calculated
using the above method and the exact numerical calculation of Hamlin
(1961).
-5-
i
1
l	
^ f
Table 1
Values of f(ao ) from Hamlin (1961) and Present Calculation
Mirror Latitude Hamlin Present
Am (degrees) Values Values
0 0.7405 0.740
4• 0.7449 0.742
8 0.7601 0.757
12 0.7842 0.781
16 0.8159 0.x'12
20 0.8535 0.849
25 0.9063 0.902
30 0.9626 0.956
35 1.020 1.014
40 1.076 1.068
45 1.129 1.119
50 1.179 1.169
55 1.224 1.213
60 1.264 1.252
-6-
The ratio Tm/Te can be computed for given mirror latitudes, am,
by calculating the two integrals in equation (2) to determine n, the
time averaged neutral hydrogen density, and using equations (1) and
(7). The calculated ratios are given in Table 2 for a m between 00
and 600 , and the corresponding equatorial pitch angle, a e , is also
presented. The temperature at ; he exobase was taken to be T c = 10000K,
the critical satellite particle altitude was taken to be R sc = 2.5R c,
and the L-value of the mirroring particles was L = 4. R c is the geo-
centric distance of the exobase taken at an altitude of 500 km. A
least squares fit of the calculated ratio, Tm/Te , to a function of the
form cos J am determined a value of J = 3.4. This is in contrast to the
previously used (Liemohn, 1961) function cos 6am'
The primary cause of the difference in the vale of the exponent
of the cosine function is the difference in the neutral hydrogen density
distribution between the Johnson and Fish (1960) model which Liemohn
(1961) used and the Chamberlain (1963) model which we have employed.
In the Johnson and Fish (1960) model the hydrogen density talls off
more rapidly as a function of radial distance than it does in the
Chamberlain (1963) model (Figure 2).
In Figure 3 we show the calculated ratio of lifetimes compared to
the fitted function cos 3 ' 4am , and compared to the previously used (Liemohn,
1961) function cos6 Xm. As can be seen from Figure 3, the function cos3.4XM
is a good approximation to the calculated values, The difference between
the two functional approximations is not as great for latitudes near 00,
corresponding to near 90 0 equatorial pitch angle, as it is at the larger
mirror latitudes. The difference at 400 mirror latitude (21.6 0 equatorial
pitch angle), however, represents nearly a -factor of 2 increase in the
-7-
i
I	 r
t	 I	 ^^	 I	 1	 '
Table 2
Ratios of Charge Exchange Lifetimes for Mirroring Particles to Equatorial
Particles for Various Mirror Latitudes
Mirror
Latitude
Am (degrees)
0.
5.
10.
15.
20.
25.
30.
35.
40.
45.
50.
55.
60.
Equatorial
Ratio Pitch angle
T	 Te. ae (degrees).
1. 90,
0.988 79.44
0.954 69.17
0.898 59.42
0.823 50.32
0.731 41.97
0.628 34.38
0.519 27.58
0.411 21.56
0.309 16.33
0.218 11.89
0.142 8.24
0.084 5.34
-8-
'	 P
charge exchange lifetimes for ions mir-r iring at that latitude.
As we stated earlier the hydrogen density in the Chamberlain
model depends on three parameters: density, NO at the exobase, tem-
perature Tc , at the exobase and the critical satellite particle altitude,
Rsc . These parameters in turn vary with the geophysical conditions
(solar activity, magnetic activity, season, etc.), and the 11SIS model
of Hedin (private communication, 1976) uses the measured geophysical
conditions to determine N c and Tc . We have considered the effect on our
functional approximation, Tm/T c = cos i xm , of varying these three para-
meters.
1) The ratio Tm /T e is independent of Nc but the actual lifetime,
T, does depend on the density at the exobase.
2) The ratio of lifetimes does depend weakly on T c . The variation
in Tm /Te for a temperature variation of 900°K to 1200 0 K is
shown in Figure 3 at point (b) for am = 30°. It should be
noted that, for the major geomagnetic storm periods during the
early life of the Explorer 45 (S 3-A) satellite ( Smith and Hoffman,
1974), Hedin's MSIS model (private communication, 1976) determines
Tc to fall in the range 9700 K to 10500K. In computing the ratios
presented in Table 2, Tc = 10000 K was used.
3) The recent observations of the Apollo 16 Lyman alpha measurements
are adequately described by a value of R sc = 2.5Rc
 particularly
in the region near 3RE ( Carruthers, 1976). However, for the
purposes of examining the effect on Tm/Te we have considered
a range of Rsc values. The variation in Tm/Te for Rsc between
-9-
I I
RIMMM
2.25Rc and 2.75Rc is shown in Figure 3 at point (c) for Xm = 400.
The effects on Tm/Te of either a Tc variation (9000K to 12000K)
or an Rsc variation (2.25Rc to 2.75Rc ) are quite small. We have computed
Tm/Te for different combinations of Tc and R sc in the stated ranges
and the maximum variations in the ratio are shown in Figure 3 at points
(a) for Xm = 200
 and am = 500 . The ratio Tm/Te was again fitted to the
cog j xm
 function, with the points weighted inversely by the spread in
Tm/Te at each Xm , and j = 3.5 was determined. In fitting the ratio
using various extreme combinations we found, for T c between 9000 K and
12000K, for Rsc between 2.25Rc and 2.75Rc and for L-values between 2.5
and 10, that j falls in the range j = 3.5 ± 0.2. Recall, that for
Tc = 10000K, Rsc = 2.5Rc and L = 4, the functional approximation was
Tm = T  cos3.4am
his new determination of the dependence of the charge exchange
lifetimes • on the mirror latitude for ions mirroring off the geomagnetic
equator requires that previous agreements or disagreements of the charge
exchange mechanism with satellite observations need to be re-examined
using at least the general relation
T = T cos3.5 t 0.2
m	 e	 m
-10-
I
"1".
I	 f	 .
Acknowledgement
We wish to thank R. E. Hartle, R. A. Hoffman, and D. P. Stern
for their very helpfu l aiscussions on this letter.
-11-
1
REFERENCES	 j
Abramowitz, M., and J. A. Stegun, Handbook of Mathematical 'Functions,
National Bureau of Standards Applied Mathematics Series, U.S.
Government Printing Office, Washington, D. C., 1965.
Bertaux, J. L., and J. E. Blamont, Interpretation of OGO-5 Lyman alpha
measurements in the upper geocorona, J. Geophys. Res., 78, 80-91,
1973.
Carruthers, G. R., T. Page, and R. R. Meier, Apollo 16 Lyman alpha
imagery of the hydrogen geocorona, J. Geo h s. Res., 81, 1664-1672,
1976.
Chamberlain, J. W., Planetary coronal and atmospheric evaporation,
Planet. Space Sci., 11, 901, 1963.
Dessler, A. J., and E. N. Parker, Hydromagnetic theory of geometric
storms, J. Geophys. Res., 64, 2239, 1959.
frar.m-, L. A., On the extraterrestrial ring current during geomagnetic
storms, J. Geophys. Res., 72, 3753-3767, 1967.
Hamlin,.D. A., R. Karplus, R. C. Vik, and K. M. Watson, Mirror and
ozimuthal drift frequencies for geomagnetically typed particles,
J. Geophys. Res., 66, 1-4, 1961.
Johnson, F. S., and R. A. Fish, The telluric hydrogen corona, Astrophys.
J., 131, 502-515, 1960.
Krylov, V. I., Approximate Calculations of Integrals, MacMillan Company,
New York, N. Y., 1962.
-12-
f i	 I
6Liemohn, N., The lifetimes of radiation beiti protons wicn energies
between 1 KeV and 1 MeV, J. Geophys. Res., 66, 3593-3595, 1961.
Lyons, L. R., and D. S. Evans, The inconsistency between proton charge
exchange and the observed ring current decay, SEL preprint #244, 1976.
McClure, G. W., Electron transfer in proton-hydrogen-atom collisions:
2-117 KeV, Phys. Rev., 148, 47-54, 1966.
Meier, R. R., and P. Mange, Geocoronal hydrogen: An analysis of the
Lyman alpha airglow observed from OGO-4, Planet. Space Sci., 18,
803-821, 1970,
Meier, R. R., and P. Mange, Spatial and temporal variations of the
Lyman alpha airglow and related atomic hydrogen distributions,
Planet. Space Sci., 21, 304-327, 1973.
Opik, E. J., and S. F. Singer, Distribution of density in a planetary
exosphere II, Phys. Fluids, 4, 221, 1961.
Prolss, G. W., Decay of the magnetic storm ring current by the charge-
exchangd mechanism, Planet. Space Sci., 21, 983-992, 1973.
Roederer, J. G., Dynamics of Geomagnetically trapped radiation, Physics
and Chemistry in Space, Vol. 2, New York, Springer-Verlag, 1970.
Russell, C. T., and R. M. Thorne, On the structure of the inner magneto-
sphere, Cosmic Electrodynamics, 1, 67-89, 1970.
-13-
4 E ,
Smith, P. H., and R. A. Hoffman, Direct observations in the dusk hours
,
of the characteristics of the storm time ring current particles g
during the beginning of magnetic storms, J. Geoophys. Res., 79,
j
966-971, 1974.
Smith, P. H., R. A. Hoffman, and T. A. Fritz, Ring current proton
	 a
decay by charge exchange, J. Geophys. Res., 81, (in press), 1976.
Y
T
Swisher, R. L., and L. A. Frank, Lifetimes for low-energy protons in 	 !
3
the outer radiation zone, J. Geophys, Res., 73, 5665-5672, 1968.
Tinsley, B. A., Evidence that the recovery phase ring current consists
of helium ions, University of Texas at Dallas preprint, 1976.
1
FIGURE CAPTIONS
Figure 1.	 Cross section of earth dipole field in a magnetic meridian
plane.
Figure 2.	 Functional dependence of atomic hydrogen density on the
radial distance from center of earth as given by the
Johnson and Fish model (shown as solid line) and the
.t
Chamberlain model (shown as dashed line). Values of
Tc = 12000K and Rsc = 2.5Rc are used. A unit density is
assumed for each model at the exobase (500km).
Figure 3.	 Ratios of lifetimes of mirroring particles to equatorial
particles as a function of mirror latitude (and of equatorial
pitch angle). Values shown by the x's correspond to input
parameters of Tc = 10000K and Rsc = 2.5Rc	in the Chamberlain
model. The approximate functional fit, cos 3.4am , is shown
by the solid line. For comparison, the function corresponding
to cos 6am (Liemohn, 1961) is shown by the dashed line.
Variations in the ratio for ranges of R sc between 2.25Rc and
2.75Rc and Tc from 9000 K to 12000 K are shown by points (a)
(am = 200 and 500 ). Variations in the ratios are shown for
fixed Rsc = 2.5Rc and Tc ranging from 9000K to 12000K at
,
point (b), and for fixed Tc = 10000 K and Rsc ranging from
2.25Rc to 2.75Rc at point (c).
-15-
F4^^4
I ;
0.0001 I	 1	 12	 3	 4
I
1.0000
0.1000
0
r
0
0.0100
3:
M
E
a
Z 0.0010
Johnson & Fish
Chamberlain
C
Geocentric Distance in RE
IJ^1.6 (b)6
......	
.
;.
EQUATORIAL PITCH ANGLE, a, (DEGREES)
(3)
E
Sz
11 1-I F
\	 COSA•A Am
\	 W
EAS E Am'^\
\
♦
♦ 	 F
I	 I	 I	 I	 I	 ,.^
m	 in	 •n	 .e	 cn	 in	 m
MIRROR LATITUDE, Am (DEGREES)
i	 ^I
}


Dependence of the charge exchange lifetimes on mirror latitude

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