- Email: [email protected]
A note on auroral electrons
Planet. sp8a
sci. w69,
Vol. 17, pp. 1051 to 1055.
Perrunon
Press. Wnted
in Northern Ireland
RESEARCH NOTES
A NOTE ON AURORAL
ELECTRONS
(Received in /inal form 22 October 1968) 1. INTRODUCTION The magnetic field inside the
tosphere is a superposition of the intrinsic field of the Earth, which is approximately that of a dipole, mYe an the secondary magnetic field produced by the currents flowing on the surface of the magnetopause. The latter has been evaluated by Mead(*)who Gnds that the total iield within the geomagnetic cavity may be represented as B=-gradV,
(1)
where V = %glo cos 0 + a 5
(r/a)“T.(fl, 9%)
(24
n-1
T,, = 5 (g,ncosm+ + km sin m+)P,,~(cos0).
(2b)
m=O The
coefficient g,O is related to the magnetic moment M of the dipole field by means of the relation _a”&” = M = 0.31.
(3) The numerical values of the other coeflicients g,n and 5,,N have been published by Mead, but with the exception of RI0and&l, these coeflicients are generally very small. The above field corresponds to a model of a magnetosphere which is approximately hemispherical on the sunlit side of the Earth and is elongated on the night side, but it probably does not take account of the neutral sheet in the tail. In two previous papers,* the nature of the trapping regions for charged particles moving in the geemagnetic field has been studied. The starting point of these investigations is the equation of motion of a charged particle in a magnetic field, namely dv Ze m--=cvhB dt where m and Z, denote respectively the mass and charge of the particle; further, the magnetic field B is given by the Equations (2a) and (2b). Such a treatment brings out all the important physical properties of the motion of a charged particle in the re@on of the radiation belts, such as the tion about a line of force and the bouncing between the mirror pomts and the global drift. The merit oPathe Stbrmer method chiefly lies in the fact that it is valid in ‘ens of very low magnetic field on the nipt side where the Alfv&n treatment is likely to break down andreFurther, it enables one to define the trappmg regions. It was shown in II that the allowed regions are determined by a cubic equation, which may be written as P + (sin A ~0s I)? + r[-2y,
+ y+)b* + yP)b*] + 1’ cos*J = 0
where r is the radial distance and L is the latitude; further b’=
ZeM d+ pe; Rs=sinA I I
and
* These will be referred to as I and II respectively. 1051
a = (-&o/O*31).
(9
1052
RESEARCH NOTES
The expressions for cpo)and yd) are lengthy and are given in II. The radial distribution of the charged particles in the region of the Van Allen radiation behs was determined in the earlier papers by obtaining the real positive root of Equation (5). In the present note, we have extended the numerical calculations to the geomagnetic tail for auroral electrons in the energy range I-50 keV. 2. THR STORMRR CONSTANT 2y, Before proceeding with the numerical results, we first discuss the physical significance of the integration constant 2y,. Tt was shown earlier that the constant 2y, is of the order of &, where rb is the radius of the magnetosphere in a spherical magnetosphere model and is otherwise a parameter of the order of 10 Earth radii. We can therefore write 2% - F
(6)
where m is a number. In a spherical magnetosphere model again, the equation of a line r = p sin* 9 1 - (af2)P
offorce is given by (7)
where p = (r&n); it follows that one can interpret the parameter m 8s the ratio of the radius of the magnetosphere to the equatorial radial distance of the line of force. If we use a cylindrical system of coordinates (R, 4, z), the integral of the azimuthal component of the equation of motion (4) for an electron has the form R’ -=d+ ds
(
bT
- ; PIP) + 2y*
(8)
It is well known that in St&mer’s theory’*’ (a i=~0), (2~& represents the com~nent (6) of the angular momentum of the electron about the geomagnetic axis, at ir&nity. In Equation (8) also, if the term inside the bracket on the right-hand side vanishes (i.e.) if a*” -_=I 2
or
r-1’26rb,
(9)
we can roughly interpret (2ydp) as the z-component of the angular momentum of the electron just near the m~etosphe~c boundary before it is injected into the cavity. The above equation yields a solution for r outside the cavity. However, if one uses a more realistic model of the magnetosphere and the full expression (1) and (2) to represent the magnetic field, the value of r is pushed very close to the magnetopause. In this case, the equation determining the allowed regions is given bY R&= ze R”_q,,+tpl’+tp”’ f2& ds ( PC “)[ r’
Thus (2y,p) can be interpreted as the z~rn~~t surface given by
1
(10)
of the angular mom~tum of an electron at points on the
-R’ _ f RP + ,/,“I + ,#*’ = 0. P
(11)
Numerical calculations have shown that the above equation gives solutions for r very close. to the magneto-
pause on the dayside and also admits real solutions witbin the etosphere on the tail side. In order to see the physical meaning of Equation (ll), let us,“fp fo owing Stem,‘6*e)represent the magnetic field by an E%&rian set of coordinates (a, Bf so that B=Va
X Vj!L
(12)
The equation of motion of a charged particle becomes dv Ze =~vx(VaxVs)-“Ft
- $
[(v . V@Vcc- (v . Va)Vgl.
(13)
A Stormer integral generally exists only for axially symmetric fields. The Equation (10) was derived by i oring small product terms of the type &,~&P’ and in the spirit of this approximation, we shall now set B = 9. Then (14)
RESEARCH
1053
NOTES
Since the magnitude of the velocity is a constant,
where $ is a unit vector in the direction of Q increasing. the $-component of Equation (13) can be written as
=
(Zelpc)[u+@a)+ - v . Va]
= -(Ze/pc)
da ;i~ + OhW
(15b)
since the first term in the square bracket in (Isa) is small, being the product of two small factors. (15b) can be integrated immediately and one obtains
R&=_
(16)
ds
A comparison shows that
Equation
of (16) and (10) brings out the relationship between Stern’s equations and our equations, and a
--=M
R' - f
R”
+
,+,y’l’ +
,,,‘“_
P
The expression on the righthand side agrees with the expression for a = a(l) given by Stern, which is correct to terms of the order &I. Equation (10) is valid within the magnetosphere as it is based on the expression (1) for the field within the cavity. However, the particles within the magnetosphere are generally of solar origin and it will be useful, if possible, to correlate the angular momentum values of the particles with their corresponding values in the solar wind just before their injection into the cavity. If we neglect the feeble fluctuating interplanetary field, the charged particles in the solar wind are essentially moving in free space and the right hand side of (4) vanishes in the equation of motion of any particle. The z-component of the equation of motion becomes Rsd$ = wrist. = 2~~. This equation is valid outside the magnetosphere but not on its boundary, with the exception of the neutral points where the field again vanishes. If we assume that the electrons and protons of the solar wind enter the cavity through the neutral points, it is possible to match the value of the constants 27, and 2y, in Equations (18) and (10). A wmparison of these two equations show that the constant 2~~ can be interpreted as the z-component (M,) of the angular momentum on the surface a = 0 and if this surface passes through the neutral points, as it is likely on the dayside, then 27. is the value of MS just before the injection of the particle into the magnetosphere through a neutral point. Numerical calculations were made to obtain the real solution of the equation a = 0 for three different longitudes 180”, 170” and 160”, on the night side and curves were obtained showing the variation of the radial distance as a function of the latitude. Figure 1 reproduces a typical curve for 4 = 180”, and the curves for the other two longitudes are almost identical. It can be seen that the curve for the radial distance intersects the equatorial plane at a point between 11 and 12 Earth radii, and this agrees with the observed location of the ne&al sheei of plasmi on the nightside, in spite of the limitations-of the Mead Geld. The radial distance is of the order of 17-20 E!arth radii for hieher latitudes and it is nrobable that the curves for a = 0 can intersect the magnetopause on the nightside. f this case these pointsAof intersection can constitute weak regions through which the solar plasma can get into the magnetosphere on the nightside. If we consider two particles of the same energy in a given element of volume in the cavity with d&rent values y,“’ and y .(*) of the integration constant, it follows from Equation (10) thatp[2y,(” - 2y,(*)] denotes the difference in the M. values of the particles. It follows from the foregoing discussion that the number m in Equation (6) is a parameter classifying the particles in accordance with the value of their z-component of the angular momentum. 3. NIJMERIcAL RESULTS Using a FerrantiShius digital computer, we have studied numerically the real positive root of Equation (5) for different values of m for electrons in the energy range l-50 keV, which corresponds to the energy spectrum of the auroral electrons. As we have stated earlier, the location of the trapping regions for particles falling in any spectrum of energy values depends signiiicantly on the value of m. Since our purpose was to understand the nature of electrons in the auroral xone, we looked for trapping regions that intersect the surface of the Earth between 70” and 90”. The numerical data show that kilovolt electrons having m values
FIO.
1.
VARIATION OF
R WITH 1
ON THE SURFACE a = 0.
approximately in the range (0,l) are confined in trapping regions that intersect the surface of the Earth in the auroral zone. Thus one can surmise that the aurora1 electrons are characterized by very low m values lying probably in the range (0,l). Also, another interesting property of these particles revealed by the numerical calculations is that they roam very wildly on the nightside of the cavity coverin fantastic distances of the order of 40-200 Earth radii at lower altitudes. This conclusion agrees well wr*& the view that the magnetosnheric tail could be a source region of auroral narticles.(7~ %ume&al tables were prepared for el&trons in the energy range l-50 keV and for the following values of m: 0.5.0.45. W4.0.35 . . . @1 and 0.05. The conclusions mentioned below could be reached from the tables : (1) Variation*of the radial distance with sin A is negligible. (2) Radial distance is not sensitive to changes in the energy of an electron in the above-mentioned range of energies. (3). Radial distance shows significant variation with respect to longitude in the equatorial region for the n@&s;ts.f$rr near the r_ lar regions, the variation of r with longitude is not entirely negligible. For = O-4and - 60”, the difference in the radial distance between the noints d = 0 and d = 180” is equal to 0.43 Earth radii. This difference becomes considerably larger fo; smaller values of m. Thus. when m = @05. the difference in the radial distance between the wints Q = 0 and & = 180” is 54 Earth radii at 1 = 75”.and 1.33 E!arth radii for the latitude il = 80”. a ’ The latitude 1, at which r = 1, depends significantly on the value of m. Figure 2 shows the variation of 1, with m as m varies from 0.05 to 0.5. A simple expression for rll can be derived from Equation (8) also bywritingr=1+rland1=?r/2-0where0issma.L Onethengets e*(1 - ;) - nl
(19)
r1 = (m/r& + (3a/2))8*’ The value of 0 corresponding to rl = 0, which we denote by O1is given by 01 = z/Q&.)
(20)
as a is small compared to unity. The above formula fits in the numerical data remarkably well. The important conclusion arrived at in this note is that the auroral electrons are characterired by very low values of the parameter m, when compared with the electrons in the Van Allen radiation belts. This
RESEARCH
NOTES
1055
so-
85 x-
f 80 -
no I
05
01 8
oi5
020
025 1
050 I
o-35
040
04?Th
o-55 !
-m Ra.2.
VARIATIONo~&wm~m.
result can also be testedexpxhnentall~. Kilovolt electrons, with small values of m or M. should be inieeted through a satellite near thk neutral points on the boundary of the magnetopause on tie sunlitside & at a distance of about 50 Barth radii in the eauatorial nlane on the nit&side of the Barth. and a eorresnondine inemase in auroral activity should be lodked for. ‘If an hruease H the intensity of the auroral e&trons i‘ observed, it provides clear evidence for the conclusion that the auroral partides are diitiqukhable from the other electrons inside the magnetosphere by their low m values. Ackwvle&eme&-The authors’ thanks are due to the Director, National Aeronautical Laboratory for permission to publish this paper and to the referee for valuable suggestions. K. S. VISWANATIUN N. mAPPA Nathal AeronauticaI Laboratory Bangaiore-17, India
REFERENCES 1. K. S. V~~WANA~HAN and P. VENKATARANCMN P&met.Spe Sci. 14,641 (1966). 2. N. RAJAPPA,P. VBNKATARANOAN and K. S. V~~WANAWAN, Phmet. Space Sci. 15,495 (1967).
3. G. D. MJUD, Lgeophys. Res. 69,118l (1964). 4. C. STORMBR, i%e Polar Aurora, Oxford University Press, London (1955).
5. D. P. STBRN,J. geophys. Res. 72, 3995 (1967). 6. D. P. STERN,Planet. Space Sci. 15, 1525 (1967). 7. BILLYM. MC&WAC, Aurora and Air&low. Proc. of the NATO Advanced Study Institute, p. 654 (1966).
sci. w69,
Vol. 17, pp. 1051 to 1055.
Perrunon
Press. Wnted
in Northern Ireland
RESEARCH NOTES
A NOTE ON AURORAL
ELECTRONS
(Received in /inal form 22 October 1968) 1. INTRODUCTION The magnetic field inside the
tosphere is a superposition of the intrinsic field of the Earth, which is approximately that of a dipole, mYe an the secondary magnetic field produced by the currents flowing on the surface of the magnetopause. The latter has been evaluated by Mead(*)who Gnds that the total iield within the geomagnetic cavity may be represented as B=-gradV,
(1)
where V = %glo cos 0 + a 5
(r/a)“T.(fl, 9%)
(24
n-1
T,, = 5 (g,ncosm+ + km sin m+)P,,~(cos0).
(2b)
m=O The
coefficient g,O is related to the magnetic moment M of the dipole field by means of the relation _a”&” = M = 0.31.
(3) The numerical values of the other coeflicients g,n and 5,,N have been published by Mead, but with the exception of RI0and&l, these coeflicients are generally very small. The above field corresponds to a model of a magnetosphere which is approximately hemispherical on the sunlit side of the Earth and is elongated on the night side, but it probably does not take account of the neutral sheet in the tail. In two previous papers,* the nature of the trapping regions for charged particles moving in the geemagnetic field has been studied. The starting point of these investigations is the equation of motion of a charged particle in a magnetic field, namely dv Ze m--=cvhB dt where m and Z, denote respectively the mass and charge of the particle; further, the magnetic field B is given by the Equations (2a) and (2b). Such a treatment brings out all the important physical properties of the motion of a charged particle in the re@on of the radiation belts, such as the tion about a line of force and the bouncing between the mirror pomts and the global drift. The merit oPathe Stbrmer method chiefly lies in the fact that it is valid in ‘ens of very low magnetic field on the nipt side where the Alfv&n treatment is likely to break down andreFurther, it enables one to define the trappmg regions. It was shown in II that the allowed regions are determined by a cubic equation, which may be written as P + (sin A ~0s I)? + r[-2y,
+ y+)b* + yP)b*] + 1’ cos*J = 0
where r is the radial distance and L is the latitude; further b’=
ZeM d+ pe; Rs=sinA I I
and
* These will be referred to as I and II respectively. 1051
a = (-&o/O*31).
(9
1052
RESEARCH NOTES
The expressions for cpo)and yd) are lengthy and are given in II. The radial distribution of the charged particles in the region of the Van Allen radiation behs was determined in the earlier papers by obtaining the real positive root of Equation (5). In the present note, we have extended the numerical calculations to the geomagnetic tail for auroral electrons in the energy range I-50 keV. 2. THR STORMRR CONSTANT 2y, Before proceeding with the numerical results, we first discuss the physical significance of the integration constant 2y,. Tt was shown earlier that the constant 2y, is of the order of &, where rb is the radius of the magnetosphere in a spherical magnetosphere model and is otherwise a parameter of the order of 10 Earth radii. We can therefore write 2% - F
(6)
where m is a number. In a spherical magnetosphere model again, the equation of a line r = p sin* 9 1 - (af2)P
offorce is given by (7)
where p = (r&n); it follows that one can interpret the parameter m 8s the ratio of the radius of the magnetosphere to the equatorial radial distance of the line of force. If we use a cylindrical system of coordinates (R, 4, z), the integral of the azimuthal component of the equation of motion (4) for an electron has the form R’ -=d+ ds
(
bT
- ; PIP) + 2y*
(8)
It is well known that in St&mer’s theory’*’ (a i=~0), (2~& represents the com~nent (6) of the angular momentum of the electron about the geomagnetic axis, at ir&nity. In Equation (8) also, if the term inside the bracket on the right-hand side vanishes (i.e.) if a*” -_=I 2
or
r-1’26rb,
(9)
we can roughly interpret (2ydp) as the z-component of the angular momentum of the electron just near the m~etosphe~c boundary before it is injected into the cavity. The above equation yields a solution for r outside the cavity. However, if one uses a more realistic model of the magnetosphere and the full expression (1) and (2) to represent the magnetic field, the value of r is pushed very close to the magnetopause. In this case, the equation determining the allowed regions is given bY R&= ze R”_q,,+tpl’+tp”’ f2& ds ( PC “)[ r’
Thus (2y,p) can be interpreted as the z~rn~~t surface given by
1
(10)
of the angular mom~tum of an electron at points on the
-R’ _ f RP + ,/,“I + ,#*’ = 0. P
(11)
Numerical calculations have shown that the above equation gives solutions for r very close. to the magneto-
pause on the dayside and also admits real solutions witbin the etosphere on the tail side. In order to see the physical meaning of Equation (ll), let us,“fp fo owing Stem,‘6*e)represent the magnetic field by an E%&rian set of coordinates (a, Bf so that B=Va
X Vj!L
(12)
The equation of motion of a charged particle becomes dv Ze =~vx(VaxVs)-“Ft
- $
[(v . V@Vcc- (v . Va)Vgl.
(13)
A Stormer integral generally exists only for axially symmetric fields. The Equation (10) was derived by i oring small product terms of the type &,~&P’ and in the spirit of this approximation, we shall now set B = 9. Then (14)
RESEARCH
1053
NOTES
Since the magnitude of the velocity is a constant,
where $ is a unit vector in the direction of Q increasing. the $-component of Equation (13) can be written as
=
(Zelpc)[u+@a)+ - v . Va]
= -(Ze/pc)
da ;i~ + OhW
(15b)
since the first term in the square bracket in (Isa) is small, being the product of two small factors. (15b) can be integrated immediately and one obtains
R&=_
(16)
ds
A comparison shows that
Equation
of (16) and (10) brings out the relationship between Stern’s equations and our equations, and a
--=M
R' - f
R”
+
,+,y’l’ +
,,,‘“_
P
The expression on the righthand side agrees with the expression for a = a(l) given by Stern, which is correct to terms of the order &I. Equation (10) is valid within the magnetosphere as it is based on the expression (1) for the field within the cavity. However, the particles within the magnetosphere are generally of solar origin and it will be useful, if possible, to correlate the angular momentum values of the particles with their corresponding values in the solar wind just before their injection into the cavity. If we neglect the feeble fluctuating interplanetary field, the charged particles in the solar wind are essentially moving in free space and the right hand side of (4) vanishes in the equation of motion of any particle. The z-component of the equation of motion becomes Rsd$ = wrist. = 2~~. This equation is valid outside the magnetosphere but not on its boundary, with the exception of the neutral points where the field again vanishes. If we assume that the electrons and protons of the solar wind enter the cavity through the neutral points, it is possible to match the value of the constants 27, and 2y, in Equations (18) and (10). A wmparison of these two equations show that the constant 2~~ can be interpreted as the z-component (M,) of the angular momentum on the surface a = 0 and if this surface passes through the neutral points, as it is likely on the dayside, then 27. is the value of MS just before the injection of the particle into the magnetosphere through a neutral point. Numerical calculations were made to obtain the real solution of the equation a = 0 for three different longitudes 180”, 170” and 160”, on the night side and curves were obtained showing the variation of the radial distance as a function of the latitude. Figure 1 reproduces a typical curve for 4 = 180”, and the curves for the other two longitudes are almost identical. It can be seen that the curve for the radial distance intersects the equatorial plane at a point between 11 and 12 Earth radii, and this agrees with the observed location of the ne&al sheei of plasmi on the nightside, in spite of the limitations-of the Mead Geld. The radial distance is of the order of 17-20 E!arth radii for hieher latitudes and it is nrobable that the curves for a = 0 can intersect the magnetopause on the nightside. f this case these pointsAof intersection can constitute weak regions through which the solar plasma can get into the magnetosphere on the nightside. If we consider two particles of the same energy in a given element of volume in the cavity with d&rent values y,“’ and y .(*) of the integration constant, it follows from Equation (10) thatp[2y,(” - 2y,(*)] denotes the difference in the M. values of the particles. It follows from the foregoing discussion that the number m in Equation (6) is a parameter classifying the particles in accordance with the value of their z-component of the angular momentum. 3. NIJMERIcAL RESULTS Using a FerrantiShius digital computer, we have studied numerically the real positive root of Equation (5) for different values of m for electrons in the energy range l-50 keV, which corresponds to the energy spectrum of the auroral electrons. As we have stated earlier, the location of the trapping regions for particles falling in any spectrum of energy values depends signiiicantly on the value of m. Since our purpose was to understand the nature of electrons in the auroral xone, we looked for trapping regions that intersect the surface of the Earth between 70” and 90”. The numerical data show that kilovolt electrons having m values
FIO.
1.
VARIATION OF
R WITH 1
ON THE SURFACE a = 0.
approximately in the range (0,l) are confined in trapping regions that intersect the surface of the Earth in the auroral zone. Thus one can surmise that the aurora1 electrons are characterized by very low m values lying probably in the range (0,l). Also, another interesting property of these particles revealed by the numerical calculations is that they roam very wildly on the nightside of the cavity coverin fantastic distances of the order of 40-200 Earth radii at lower altitudes. This conclusion agrees well wr*& the view that the magnetosnheric tail could be a source region of auroral narticles.(7~ %ume&al tables were prepared for el&trons in the energy range l-50 keV and for the following values of m: 0.5.0.45. W4.0.35 . . . @1 and 0.05. The conclusions mentioned below could be reached from the tables : (1) Variation*of the radial distance with sin A is negligible. (2) Radial distance is not sensitive to changes in the energy of an electron in the above-mentioned range of energies. (3). Radial distance shows significant variation with respect to longitude in the equatorial region for the n@&s;ts.f$rr near the r_ lar regions, the variation of r with longitude is not entirely negligible. For = O-4and - 60”, the difference in the radial distance between the noints d = 0 and d = 180” is equal to 0.43 Earth radii. This difference becomes considerably larger fo; smaller values of m. Thus. when m = @05. the difference in the radial distance between the wints Q = 0 and & = 180” is 54 Earth radii at 1 = 75”.and 1.33 E!arth radii for the latitude il = 80”. a ’ The latitude 1, at which r = 1, depends significantly on the value of m. Figure 2 shows the variation of 1, with m as m varies from 0.05 to 0.5. A simple expression for rll can be derived from Equation (8) also bywritingr=1+rland1=?r/2-0where0issma.L Onethengets e*(1 - ;) - nl
(19)
r1 = (m/r& + (3a/2))8*’ The value of 0 corresponding to rl = 0, which we denote by O1is given by 01 = z/Q&.)
(20)
as a is small compared to unity. The above formula fits in the numerical data remarkably well. The important conclusion arrived at in this note is that the auroral electrons are characterired by very low values of the parameter m, when compared with the electrons in the Van Allen radiation belts. This
RESEARCH
NOTES
1055
so-
85 x-
f 80 -
no I
05
01 8
oi5
020
025 1
050 I
o-35
040
04?Th
o-55 !
-m Ra.2.
VARIATIONo~&wm~m.
result can also be testedexpxhnentall~. Kilovolt electrons, with small values of m or M. should be inieeted through a satellite near thk neutral points on the boundary of the magnetopause on tie sunlitside & at a distance of about 50 Barth radii in the eauatorial nlane on the nit&side of the Barth. and a eorresnondine inemase in auroral activity should be lodked for. ‘If an hruease H the intensity of the auroral e&trons i‘ observed, it provides clear evidence for the conclusion that the auroral partides are diitiqukhable from the other electrons inside the magnetosphere by their low m values. Ackwvle&eme&-The authors’ thanks are due to the Director, National Aeronautical Laboratory for permission to publish this paper and to the referee for valuable suggestions. K. S. VISWANATIUN N. mAPPA Nathal AeronauticaI Laboratory Bangaiore-17, India
REFERENCES 1. K. S. V~~WANA~HAN and P. VENKATARANCMN P&met.Spe Sci. 14,641 (1966). 2. N. RAJAPPA,P. VBNKATARANOAN and K. S. V~~WANAWAN, Phmet. Space Sci. 15,495 (1967).
3. G. D. MJUD, Lgeophys. Res. 69,118l (1964). 4. C. STORMBR, i%e Polar Aurora, Oxford University Press, London (1955).
5. D. P. STBRN,J. geophys. Res. 72, 3995 (1967). 6. D. P. STERN,Planet. Space Sci. 15, 1525 (1967). 7. BILLYM. MC&WAC, Aurora and Air&low. Proc. of the NATO Advanced Study Institute, p. 654 (1966).