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Variations in the doppler profile of Hα in aurorae
Planet. Space Sd. 1963, Vol. 11.
pp,1229
to 1231.
VARIATIONS
Pcr~amon PreJa Ltd
Printed in Northern Ireland
IN THE DOPPLER PROFILE OF Ha IN AURORAE
0. E. JOHANSEN and A. OMI-IOLT Institute of Physics, University of Oslo, Blindern, Norway (Received 18 June 1963) Abstract-Variations in the observed Doppler protie of Ha emitted from high latitude aurorae are reported. On one particular night the half-width of the horizon protile ranged from about 15 to 23 A. The variations are interpreted as changes in the pitch angle and velocity distributions of the incident protons. With an assumed pitch angle distribution of the form cost B the index R seems to vary from below 0 to about 6. Theoretical considerations show that this may indicate that the protons are accelerated in the vicinity of the Earth.
During the last ten years much work has been devoted to the study of the intensities and Doppler pro&es of hydrogen lines in auroras. This work has been reviewed by Chamberlain(l) and by Galperin (2). Apart from the very narrow hydrogen lines which have been recorded by MontalbettP and Galperin@), few variations in the width of the lines have been observed, despite the fact that a great number of spectrograms with reasonable resolution have been obtained. The observations by Omholt, Stoffregen and Derblom(6) with a scanning spectrophotometer, indicating variations in the Ha profile in the zenith, are unique. The purpose of this paper is to report further observations with this instrument, which show conclusively that significant variations in the Ho: profile do occur. 2. THE APPEARANCE AND INTENSITY OF Ha 2.1. The observations were made in Tromsii with the ins~ent scanning the spectral range from 6250 to 6600 A in one minute so that rapid changes in the spectrum could be detected. A parallel filter photometer fitted with an entrance slit recorded any intensity variations of the aurora during the scan of the spectrum. Both instruments had field lenses which focused the aurora on the slits in such a way that each picked out the same area in the sky. This was about 6” in a horizontal direction and a fraction of a degree in a vertical direction. It was usually not possible to draw profiles with sufficient accuracy from one single scan and a few (2-14) consecutive spectra were averaged to obtain good profiles. 2.2. During the period January, February and March 1962,161 spectra were obtained from 13 different aurorae: 55 spectra were taken towards the magnetic zenith, 106 towards the horizon: 8 of the aurorae were diffuse glows, 1 diffuse, yellow-white arc (to the south) and 4 invisible to the eye. Observations thus confirm earlier suggestion that hydrogen emission at high latitude is mainly associated with diffuse glow. In no case was Ha emission observed in the stronger distinct aurora1 forms which occur sirn~~eo~ly with the glow. In some cases, but not all, there was a dark belt in between the hydrogen emitting area of the sky and the area where brighter forms appeared. The intensity of Ha was usuahy about the same as that of the red oxygen lines 63006364 A, whereas the first positive N2 bands in the same spectral range were usually too weak to be recorded. 1223
1224
0. E. JOHANSEN and A. QMHQLT
The hydrogen emission in some cases is succeeded by a brighter auroraZ display, but in about as many c135esis not. The ob~~a~~~s are in a~~~rn~~t with the evidence of appears mainly south of the brighter ~t~~~g~~ and rlerblom fs) that the hydrogen assign forms before eight and north of them after ~d~ght, but the material is nat suflkknt to prove this conclusively.
b)
2.3. ~x~p~~s of recorded profks are shown in Fig. 1, and the resxzlts are s~rn~iz~d in Tables f and 2. It is quite clear that si~~cant v~at~ons occur, The horizon profiles show very little variation except during the night of 15-16 February 1963, when large variations occur. All the profifes from this night are similar in shape, but the ham-width
VARIATIONS
IN THE DOPPLER
PROFILE
1225
OF Ha IN AURORAE
TABLE1 Zenith profiles. Averages of observed distances from the undisplaced Ha line to the maximum intensity point (A&,$, and to the huff-ma~mum intensity point at shorter (Ad& and longer (AX& wavelengths. The standard deviation c for each measured quantity is also given. For comparison the profile published by Ch~~rl~n(‘) is included. A
am A
Ah,
55 xenith profiles
6.0
2.1
21
7 profiles about 21 .I5 MET 15. February
4.9
19.6
6.4
6 profiles about 23.00 MET 15. February
7-8
21.0
4.2
ChamberlahW
4-5
20
3
A&,
A
alIf)
A
Ax 11s A
wp
6
5.1
l-9
A
TABLE 2
Horizon profiles. Averages of half-intensity widths A& (between half-of-maximum intensity points) and standard deviation u.
All except 4 sharp profiles AU except profiles from the night 15-l 6th February
Total number
A&Z A
i
102
16.7
25
30
14.7
o-3
Charn~rl~(”
15
23-
x
22 -
16 I5-
x I IS
III 20
21
22
I 23
X-X 1 I\ "l\*RX I I I I 00 01 02 03
[ 04
MET
Fm.2.
Trm HALF wx~mi A& OF THEHORIZON PROFILE OF Ha OWERVED 15-16th FEBRI_JARY~~~~,PLOT~EDAGAINSTTIME.
DURING THE NIGHT
1226
0.
E. JOHANSEN
and A. OMHOLT
Aiz, varies significantly during the night. Figure 2 shows Aiz, plotted against time. Observations towards the magnetic horizon were not taken continuously since, for much of the time, the instrument was used to search the rest of the sky for hydrogen emission. Omitting this night, the average horizon profile is in good agreement with those found earlier(l*‘). From the estimated inaccuracies in determining the quantities given in Tables 1 and 2 a standard deviation (root mean square error) of about 0.5 A is expected if no real variations in the profiles has occurred. As is seen, the standard deviation is much larger than this in the case of the zenith profiles. There are thus significant variations in the zenith profile, but these variations do not seem to be connected to typical variations in the auroral form, to the time of night or to any particular nights. During the night of February 14-15th, when the variations in the horizon profile occurred, only a few profiles were obtained in the zenith at about 21.15 and 23.00 MET, and these were close to the average. The average profile from our observations is in good agreement with those observed earlier’1~7). While there is no significant correlation between AL, and A& or between A&s and AX1,2 for the zenith profiles, there is a negative correlation between A2, and A3L;,2,partly, but not wholly, due to random errors in the fixation of the undisplaced line. The correlation may be explained by assuming that the positive tail of the zenith profile is due to protons with low energy or high pitch angle, many of which may be scattered upwards. An increase in the relative number of these would increase AA’1,2,while it would decrease AL, by displacing the maximum of the profile towards the undisplaced line. 3. INTERPRFtTATION
3.1. It seems possible to give a reasonable explanation of the average profiles observed earlier by assuming the protons are distributed with initial velocity u,, according to the inverse power law v~-~,above a certain cut-off; and by assuming that just before they enter the atmosphere their distribution with pitch angle 15is ~(0) = cos” 8, where ~(0) is the intensity in protons/cma set sterad, and the unit area is taken perpendicular to the velocity vector(l). The influx of particles per cm2 surface of the atmosphere, per set per unit velocity and per sterad, is then given by N(u,, e) = 0:
8 > 42
~(~a, e) = cv,-* ~09 e . cos 8: = N, & (S - I)(~ + 2)v;-
e < 42
(1)
ra-8 COS~+~ 8,
N, being the total influx of protons with initial velocity higher than the cut-off. velocity Vmin.
We have, as a first approach, tried to interpret the observed profiles in this way, i.e. we postulate the distribution given by (1) and determine the values of II and s which yield the best fit to the observations. 3.2. When profiles are obtained both in the magnetic zenith and the horizon from the same aurora, n and s can be obtained by methods described by Chamberlain(l). It can be shown that the angular distribution exponent n is then given by 8% -=-
fiz
2 s TO
42
COS~+~ e de,
(2)
VARIATIONS
IN THE DOPPLER
v,
v,
FXG.3,
&%%WT!D
(----)
PROFILE
1227
kmlsec
kmtsec
AND COhSPUTED (- - - - ) PRosm!s Zl?.NlT?i (U)
OF Ha IN AURORAE
AND
HORIZON
OF
Ha
IN DIRECTXONS
OF MAGNBTIC
(6).
The zenith profile is also corrected for N,-emission 6 * +*). The adopted distribution function for the protons is given by equation (1) with p1= 1.85, s = 2.97 and u,~,, < 45 km,&. The curves are normalized such that the zenith profiles have the same maximum value and the horizon profiles have the same amplitude at v = 200 km/set.
where i; and 5, are the average Doppler velocities in the directions of the magnetic horizon and the magnetic zenith, derived from the respective observed profiles@). When n has been determined from (2) and the measured values of G*and t?$,s may be determined by some other property such as the position of the maximum point of the zenith proflleP. It is not possible to obtain simultaneous horizon and zenith profiles of the same aurora from one station. The variability of the profiles, as demonstrated in $2, makes it clear that such observations are needed. Our observations for the night of 15-l 6th Feb. were almost simul~neous from parts of the aurora differing by about 5’ in latitude. From these profjles n was found to be about 1~9and s close to 3.0, using the method indicated. Detailed profiles were hence computed numerically, adopting the basic emission functions and procedure given by Cham~rl~n(l) and using an IBM 1620 computer. The cut-off velocity for the velocity distribution function was chosen to be less than 45 kmlsec.
1228
0.
E. JOHANSEN
and A. OMHOLT
The resulting detailed fit shown in Fig. 3 is not very good, which may indicate that the basic formulae adopted are imprecise. However, it should also be born in mind that the values of i& and r?, derived from the profile may be somewhat in error, because they depend much on the wings of the profiles, which may easily be obscured by other emissions. As pointed out by Chamberlain (l) the high velocity tail of Ha in the zenith may be contaminated by the first positive A$, bands and thus be overestimated. It is also quite likely that the tail of the horizon profile can be underestimated, because the higher and weaker part of the aurora, which would contribute much of this emission, may often not be included in the observed part of the aurora. These effects would lead to overestimation of n.
2000
FG 4.
ZENITH
PROFIJLE OF
1000
1500 Y,
0
SO0
kmlsec
Hu, OMNG n (Cf. EQUATION
=
0.6
&
0.2
AND
s =
2.97 & 0.02
(1)).
From profiles published earlier a value of 6 was obtained for n by the same method(V). The difference is mainly due to the broader horizon profile used in our computations. 3.3. Based on the distribution function given by (1) we have also tried to derive information on the apparent variation of la and s from the zenith profile alone. The parameters Ait, and A&,2, defined in $2, were computed numerically from the same theoretical results, and plotted as functions of n and s. By comparison with observed profiles n and s were determined. In 12 of 55 cases this method gave satisfacto~ a~eement between observed and computed profiles. These profiles could be divided in three distinctly different groups, yielding values of n about O-6,2*8 and 6-O respectively. In all cases s was close to 3. One of these profiles, giving n = O-6, is shown in Fig. 4. As is seen, also the detailed fit between the computed and measured profiles is reasonably good in these cases. All the other cases indicate values of n between -0.5 and - 1, and values of s below 2.5. As seen from (1) la = - 1 implies isotropic influx of protons. I3y symmetry arguments it is clear that in this case the zenith profile would be exactly one half of the horizon profile; thus only positive velocities would be missing from the zenith profile while they would he present in the horizon profile. Neglecting the extremely narrow profiles, it is not possible to find one horizon and one zenith profile which, combined, would fit this picture. It is therefore not realistic to accept the results indicating n close to - 1, derived from the zenith profile alone.
VARIATIONS
IN THE DOPPLER
PROFILE
OF Ha IN AURORAE
1229
When the method adopted in this section is applied to the zenith profiles used in $3.2 values of n and s are obtained which are smaller than those obtained in $3.2. This is in agreement with the indication mentioned in $3.2, that the value of n derived there may be too large. Also, one must bear in mind that the zenith and horizon profiles used in 93.2 were not from the same part of the aurora. The aurora observed in the magnetic horizon is about 5” latitude further north than that observed in the magnetic zenith, so that the proton distribution function may be very different. 4. DISCUSSION
4.1. The tentative interpretation of the observed variations in the Ha profiles given in $3 cannot be regarded as satisfactory. It seems reasonably certain however, that great and important variations occur in the angular distribution, but it is hardly realistic or justifiable to assume that variations in the observed profiles can be explained by variations in the angular distribution only, even if this departed very much from the cosn e-form. A suitable change in the distribution function could thus also make the theoretical curves fit better with the observed ones (Fig. 3). For example, a somewhat better fit than that shown in Fig. 3 would have been obtained if a velocity distribution had been used which gave relatively fewer protons at high initial velocities, u,. This could be achieved by letting s increase with increasing ZJ-,.Such a change would also allow greater values of n (cf. $3.3). At low velocities the curves cannot be regarded as reliable because of the great uncertainties in the basic data. That variations in the energy distribution occur is also indicated by rocket measurements, which yield greatly varying energy spectra of protons @glO).But since rocket observations still are confined to fairly high energies only, they are not directly comparable with the observed Ha-profiles, where protons of lower energy apparently dominate the shaping of the profile. Although the emission function for Ha is not very accurately known, much more could be learned of the variations in the angular and velocity distribution of protons if simultaneous observations were made of the zenith and horizon profiles of hydrogen lines emitted from the same part of the aurora. This is because variations in the two distributions affect the relative widths of the two profiles differently; for example, a widening of the angular distribution (decreasing n in (1)) would increase the width of the horizon profile, whereas it would decrease that of the zenith profile. On the other hand, a widening of the velocity distribution (decreasing s in (1)) would increase the width of both profiles. There is therefore an urgent need for such observations. 4.2. The adiabatic equation which governs the movement of the protons in the Earth’s magnetic field reads E sin2 0
-
B
=
E, sin2 0, Bo
’
(3)
where E is the kinetic energy of the protons and B the magnetic field(lv”). This equation is also valid if E changes because of acceleration of the protons in an electric field, unless important potential differences occur over distances comparable to the gyro-radius of the protons (and not directed along B),or unless oscillations in the electric field occur, with frequencies comparable to or higher than the gyro frequency. With the same reservations, acceleration in directions across B cannot occur because of the Hall-effect.
0. E. JOHANSEN
1230
and A. OMHOLT
The relation between the angular distribution of the total particle influx, N(e) = r,@) cos 8, at any point, compared to the reference point indexed 0, is then given by (4) where iV(e) and q(0) have the same significance as in $3.1. This may be shown directly or by Liouville’s theorem (lg8). Suppose for example, there is a reservoir of particles with MaxweIlian distribution in E,, with characteristic temperature T,, from which the protons are accelerated to energy E. The distribution of these particles with pitch angle and initiai energy is given by
(5) if we assume N(0,) to be isotropic.
r(e)
cos
e = iv(e) =
cl
By integration over E, we get EB,COS
B
8
*
i
s I,
E,, COS 6$
exp (--EoW,)
dEo,
(6)
where E,,, is the minimum allowable value of E. as given by (3): E,
=
2sins8,
(7)
(since sin2 0, I; 1) and from (3): cos e, =
1 - ;i
sin2 e
0
l/2 = (1 - E,,JEJ1’?
Substituting x = Eo/E,,, gives
v(e)
cos
e = N(e) =
cl
J-0 B
cos e
sco
exp(--JcEmlkTo)
1
dx
(X2 -
x)1/2
-
(9)
This equation shows that with increasing 8 (increasing E,,J IV(e) falls off faster than cos 8, provided E, is not very small. Moreover, a study of the derivatives of the integral in (9) with respect to i? and (EB,/B) shows that the higher EBo/B is, the more narrowly q(e) is concentrated around 8 = 0. On the other hand, if no acceleration takes place so that E = E,, then from (4) and (6) the angular distribution for a given vafue of E is given by
r(e)
cos
e = iv(e) =
cl
Bo
cos
e N(e,)
z
1 - 4 sina e i
l/2*
00)
1
But since the particles probably originate somewhere out in space where B. < B and N(0,) is nearly isotropic over the range of 0, in question, we have '17(e)~0~e = N(e) = C,COS~.
(11)
In this case the distribution function r(e) would be isotropic, regardless of energy, i.e. n = 0 in (1).
VARIATIONS
IN THE DOPPLER
PROFILE
OF Ha IN AURORAE
1231
By draining from a reservoir of trapped particles or stepwise acceleration effected by bouncing between magnetic mirrors, a wide distribution can be expected; the protons would escape and move into the atmosphere as soon as they satisfy (3) with 0 = 7r/2 and B as the value of the magnetic field just above the atmosphere. All particles with low 8 would be drained out immediately leaving only particles with high 8 when a quasisteady state is reached. In general, we may say that if EBo/B > kT,,, then there will be a pronounced fall off in ~(0) with 8, whereas if EBo/B < kT, q(O) will be nearly isotropic. The fall off will be steeper with increasing EB,,/B. The observational evidence suggests that in some cases r(O) falls off as 0 increasesindeed this may always be the case. If q(O) varies as seems likely, it clearly cannot always be isotropic. If the theory outlined above has any resemblance to reality, then this fall off in l;l(O) implies that EBo/B > kT,, where B, and To are the magnetic field and the temperature of the plasma before the protons are accelerated. The distance into space at which acceleration starts is limited if the temperature of the plasma is not low enough to ensure that this relation is fulfilled with the interplanetary value of the magnetic field for B,,. Another point is that with this model protons with higher final energies E before entry into the atmosphere would show a more narrow angular distribution than those with smaller energies. Inspection of the observed profiles may lend some support to this view. Such a variation would, for example, improve the computed horizon profile shown in Fig. 3, as the center peak (caused by slow protons with small 0) would be more spread out and the wings (caused by fast protons with high 0) would be diminished. Changes in the velocity distribution as indicated in $4.1 could then probably give the necessary correction to the computed zenith profiles. We could certainly make a number of trial and error computations with various model distributions and find some which would give a good fit to any profile. Similarly, we could make theoretical computations assuming various conditions in space. However, this hardly seems worthwhile since observational material is still meagre and unsatisfactory, and uncertainties still exist in the basic emission function for a proton-hydrogen beam in air. Nevertheless, observations of this kind have great potentiality. Acknowledgements-The work reported in this paper has been supported in part by the Cambridge Research Laboratories, OAR, through the European Office, Aerospace Research, U.S. Air Force, contract AF 61(052)252. It is also a pleasure to express our thanks to the staff at the Auroral Observatory for valuable help in various ways, and to Professor D. R. Bates, F.R.S., for helpful comments on the manuscript. REFERENCES 1. J. W. CHAMBERLAIN, Physics of the Aurora and Airglow, p. 264. Academic Press, New York and London (1961). 2. Y. I. GALPERIN,Planet. Space Sci. 10, 187 (1963). 3. R. MONTALBETII,J. Atmos. Terr. Phys. 14,200 (1959). 4. J. M. MALMLLE.,Planet. Space Sci. 2, 130 (1960). 5. A. OMHOLT,W. STOFFREGEN and H. DERBLOM,J. Atmos. Terr. Phys. 24,203 (1962). 6. W. STOFFREGEN and H. DERBLOM,Planet. Space Sci. 9,711 (1962). 7. J. W. CI-L4MBEIUArN, Sky & Telex. 17, 339 (1958). 8. A. OMHOLT,Geofys. Publ. 20, No. 11 (1959). 9. L. H. MEREDITH,L. R. DAVIS, J. P. HEPPNERand 0. BERG, Z.G. Y. Rocket Report Series. No. 1, p. 169 (1958). 10. C. E. McILw~, J. Geophys. Res. 6!5,2727 (1960). 11. H. AL&N, Cosmical Electrodynamics. Clarendon Press, Oxford (1950).
pp,1229
to 1231.
VARIATIONS
Pcr~amon PreJa Ltd
Printed in Northern Ireland
IN THE DOPPLER PROFILE OF Ha IN AURORAE
0. E. JOHANSEN and A. OMI-IOLT Institute of Physics, University of Oslo, Blindern, Norway (Received 18 June 1963) Abstract-Variations in the observed Doppler protie of Ha emitted from high latitude aurorae are reported. On one particular night the half-width of the horizon protile ranged from about 15 to 23 A. The variations are interpreted as changes in the pitch angle and velocity distributions of the incident protons. With an assumed pitch angle distribution of the form cost B the index R seems to vary from below 0 to about 6. Theoretical considerations show that this may indicate that the protons are accelerated in the vicinity of the Earth.
During the last ten years much work has been devoted to the study of the intensities and Doppler pro&es of hydrogen lines in auroras. This work has been reviewed by Chamberlain(l) and by Galperin (2). Apart from the very narrow hydrogen lines which have been recorded by MontalbettP and Galperin@), few variations in the width of the lines have been observed, despite the fact that a great number of spectrograms with reasonable resolution have been obtained. The observations by Omholt, Stoffregen and Derblom(6) with a scanning spectrophotometer, indicating variations in the Ha profile in the zenith, are unique. The purpose of this paper is to report further observations with this instrument, which show conclusively that significant variations in the Ho: profile do occur. 2. THE APPEARANCE AND INTENSITY OF Ha 2.1. The observations were made in Tromsii with the ins~ent scanning the spectral range from 6250 to 6600 A in one minute so that rapid changes in the spectrum could be detected. A parallel filter photometer fitted with an entrance slit recorded any intensity variations of the aurora during the scan of the spectrum. Both instruments had field lenses which focused the aurora on the slits in such a way that each picked out the same area in the sky. This was about 6” in a horizontal direction and a fraction of a degree in a vertical direction. It was usually not possible to draw profiles with sufficient accuracy from one single scan and a few (2-14) consecutive spectra were averaged to obtain good profiles. 2.2. During the period January, February and March 1962,161 spectra were obtained from 13 different aurorae: 55 spectra were taken towards the magnetic zenith, 106 towards the horizon: 8 of the aurorae were diffuse glows, 1 diffuse, yellow-white arc (to the south) and 4 invisible to the eye. Observations thus confirm earlier suggestion that hydrogen emission at high latitude is mainly associated with diffuse glow. In no case was Ha emission observed in the stronger distinct aurora1 forms which occur sirn~~eo~ly with the glow. In some cases, but not all, there was a dark belt in between the hydrogen emitting area of the sky and the area where brighter forms appeared. The intensity of Ha was usuahy about the same as that of the red oxygen lines 63006364 A, whereas the first positive N2 bands in the same spectral range were usually too weak to be recorded. 1223
1224
0. E. JOHANSEN and A. QMHQLT
The hydrogen emission in some cases is succeeded by a brighter auroraZ display, but in about as many c135esis not. The ob~~a~~~s are in a~~~rn~~t with the evidence of appears mainly south of the brighter ~t~~~g~~ and rlerblom fs) that the hydrogen assign forms before eight and north of them after ~d~ght, but the material is nat suflkknt to prove this conclusively.
b)
2.3. ~x~p~~s of recorded profks are shown in Fig. 1, and the resxzlts are s~rn~iz~d in Tables f and 2. It is quite clear that si~~cant v~at~ons occur, The horizon profiles show very little variation except during the night of 15-16 February 1963, when large variations occur. All the profifes from this night are similar in shape, but the ham-width
VARIATIONS
IN THE DOPPLER
PROFILE
1225
OF Ha IN AURORAE
TABLE1 Zenith profiles. Averages of observed distances from the undisplaced Ha line to the maximum intensity point (A&,$, and to the huff-ma~mum intensity point at shorter (Ad& and longer (AX& wavelengths. The standard deviation c for each measured quantity is also given. For comparison the profile published by Ch~~rl~n(‘) is included. A
am A
Ah,
55 xenith profiles
6.0
2.1
21
7 profiles about 21 .I5 MET 15. February
4.9
19.6
6.4
6 profiles about 23.00 MET 15. February
7-8
21.0
4.2
ChamberlahW
4-5
20
3
A&,
A
alIf)
A
Ax 11s A
wp
6
5.1
l-9
A
TABLE 2
Horizon profiles. Averages of half-intensity widths A& (between half-of-maximum intensity points) and standard deviation u.
All except 4 sharp profiles AU except profiles from the night 15-l 6th February
Total number
A&Z A
i
102
16.7
25
30
14.7
o-3
Charn~rl~(”
15
23-
x
22 -
16 I5-
x I IS
III 20
21
22
I 23
X-X 1 I\ "l\*RX I I I I 00 01 02 03
[ 04
MET
Fm.2.
Trm HALF wx~mi A& OF THEHORIZON PROFILE OF Ha OWERVED 15-16th FEBRI_JARY~~~~,PLOT~EDAGAINSTTIME.
DURING THE NIGHT
1226
0.
E. JOHANSEN
and A. OMHOLT
Aiz, varies significantly during the night. Figure 2 shows Aiz, plotted against time. Observations towards the magnetic horizon were not taken continuously since, for much of the time, the instrument was used to search the rest of the sky for hydrogen emission. Omitting this night, the average horizon profile is in good agreement with those found earlier(l*‘). From the estimated inaccuracies in determining the quantities given in Tables 1 and 2 a standard deviation (root mean square error) of about 0.5 A is expected if no real variations in the profiles has occurred. As is seen, the standard deviation is much larger than this in the case of the zenith profiles. There are thus significant variations in the zenith profile, but these variations do not seem to be connected to typical variations in the auroral form, to the time of night or to any particular nights. During the night of February 14-15th, when the variations in the horizon profile occurred, only a few profiles were obtained in the zenith at about 21.15 and 23.00 MET, and these were close to the average. The average profile from our observations is in good agreement with those observed earlier’1~7). While there is no significant correlation between AL, and A& or between A&s and AX1,2 for the zenith profiles, there is a negative correlation between A2, and A3L;,2,partly, but not wholly, due to random errors in the fixation of the undisplaced line. The correlation may be explained by assuming that the positive tail of the zenith profile is due to protons with low energy or high pitch angle, many of which may be scattered upwards. An increase in the relative number of these would increase AA’1,2,while it would decrease AL, by displacing the maximum of the profile towards the undisplaced line. 3. INTERPRFtTATION
3.1. It seems possible to give a reasonable explanation of the average profiles observed earlier by assuming the protons are distributed with initial velocity u,, according to the inverse power law v~-~,above a certain cut-off; and by assuming that just before they enter the atmosphere their distribution with pitch angle 15is ~(0) = cos” 8, where ~(0) is the intensity in protons/cma set sterad, and the unit area is taken perpendicular to the velocity vector(l). The influx of particles per cm2 surface of the atmosphere, per set per unit velocity and per sterad, is then given by N(u,, e) = 0:
8 > 42
~(~a, e) = cv,-* ~09 e . cos 8: = N, & (S - I)(~ + 2)v;-
e < 42
(1)
ra-8 COS~+~ 8,
N, being the total influx of protons with initial velocity higher than the cut-off. velocity Vmin.
We have, as a first approach, tried to interpret the observed profiles in this way, i.e. we postulate the distribution given by (1) and determine the values of II and s which yield the best fit to the observations. 3.2. When profiles are obtained both in the magnetic zenith and the horizon from the same aurora, n and s can be obtained by methods described by Chamberlain(l). It can be shown that the angular distribution exponent n is then given by 8% -=-
fiz
2 s TO
42
COS~+~ e de,
(2)
VARIATIONS
IN THE DOPPLER
v,
v,
FXG.3,
&%%WT!D
(----)
PROFILE
1227
kmlsec
kmtsec
AND COhSPUTED (- - - - ) PRosm!s Zl?.NlT?i (U)
OF Ha IN AURORAE
AND
HORIZON
OF
Ha
IN DIRECTXONS
OF MAGNBTIC
(6).
The zenith profile is also corrected for N,-emission 6 * +*). The adopted distribution function for the protons is given by equation (1) with p1= 1.85, s = 2.97 and u,~,, < 45 km,&. The curves are normalized such that the zenith profiles have the same maximum value and the horizon profiles have the same amplitude at v = 200 km/set.
where i; and 5, are the average Doppler velocities in the directions of the magnetic horizon and the magnetic zenith, derived from the respective observed profiles@). When n has been determined from (2) and the measured values of G*and t?$,s may be determined by some other property such as the position of the maximum point of the zenith proflleP. It is not possible to obtain simultaneous horizon and zenith profiles of the same aurora from one station. The variability of the profiles, as demonstrated in $2, makes it clear that such observations are needed. Our observations for the night of 15-l 6th Feb. were almost simul~neous from parts of the aurora differing by about 5’ in latitude. From these profjles n was found to be about 1~9and s close to 3.0, using the method indicated. Detailed profiles were hence computed numerically, adopting the basic emission functions and procedure given by Cham~rl~n(l) and using an IBM 1620 computer. The cut-off velocity for the velocity distribution function was chosen to be less than 45 kmlsec.
1228
0.
E. JOHANSEN
and A. OMHOLT
The resulting detailed fit shown in Fig. 3 is not very good, which may indicate that the basic formulae adopted are imprecise. However, it should also be born in mind that the values of i& and r?, derived from the profile may be somewhat in error, because they depend much on the wings of the profiles, which may easily be obscured by other emissions. As pointed out by Chamberlain (l) the high velocity tail of Ha in the zenith may be contaminated by the first positive A$, bands and thus be overestimated. It is also quite likely that the tail of the horizon profile can be underestimated, because the higher and weaker part of the aurora, which would contribute much of this emission, may often not be included in the observed part of the aurora. These effects would lead to overestimation of n.
2000
FG 4.
ZENITH
PROFIJLE OF
1000
1500 Y,
0
SO0
kmlsec
Hu, OMNG n (Cf. EQUATION
=
0.6
&
0.2
AND
s =
2.97 & 0.02
(1)).
From profiles published earlier a value of 6 was obtained for n by the same method(V). The difference is mainly due to the broader horizon profile used in our computations. 3.3. Based on the distribution function given by (1) we have also tried to derive information on the apparent variation of la and s from the zenith profile alone. The parameters Ait, and A&,2, defined in $2, were computed numerically from the same theoretical results, and plotted as functions of n and s. By comparison with observed profiles n and s were determined. In 12 of 55 cases this method gave satisfacto~ a~eement between observed and computed profiles. These profiles could be divided in three distinctly different groups, yielding values of n about O-6,2*8 and 6-O respectively. In all cases s was close to 3. One of these profiles, giving n = O-6, is shown in Fig. 4. As is seen, also the detailed fit between the computed and measured profiles is reasonably good in these cases. All the other cases indicate values of n between -0.5 and - 1, and values of s below 2.5. As seen from (1) la = - 1 implies isotropic influx of protons. I3y symmetry arguments it is clear that in this case the zenith profile would be exactly one half of the horizon profile; thus only positive velocities would be missing from the zenith profile while they would he present in the horizon profile. Neglecting the extremely narrow profiles, it is not possible to find one horizon and one zenith profile which, combined, would fit this picture. It is therefore not realistic to accept the results indicating n close to - 1, derived from the zenith profile alone.
VARIATIONS
IN THE DOPPLER
PROFILE
OF Ha IN AURORAE
1229
When the method adopted in this section is applied to the zenith profiles used in $3.2 values of n and s are obtained which are smaller than those obtained in $3.2. This is in agreement with the indication mentioned in $3.2, that the value of n derived there may be too large. Also, one must bear in mind that the zenith and horizon profiles used in 93.2 were not from the same part of the aurora. The aurora observed in the magnetic horizon is about 5” latitude further north than that observed in the magnetic zenith, so that the proton distribution function may be very different. 4. DISCUSSION
4.1. The tentative interpretation of the observed variations in the Ha profiles given in $3 cannot be regarded as satisfactory. It seems reasonably certain however, that great and important variations occur in the angular distribution, but it is hardly realistic or justifiable to assume that variations in the observed profiles can be explained by variations in the angular distribution only, even if this departed very much from the cosn e-form. A suitable change in the distribution function could thus also make the theoretical curves fit better with the observed ones (Fig. 3). For example, a somewhat better fit than that shown in Fig. 3 would have been obtained if a velocity distribution had been used which gave relatively fewer protons at high initial velocities, u,. This could be achieved by letting s increase with increasing ZJ-,.Such a change would also allow greater values of n (cf. $3.3). At low velocities the curves cannot be regarded as reliable because of the great uncertainties in the basic data. That variations in the energy distribution occur is also indicated by rocket measurements, which yield greatly varying energy spectra of protons @glO).But since rocket observations still are confined to fairly high energies only, they are not directly comparable with the observed Ha-profiles, where protons of lower energy apparently dominate the shaping of the profile. Although the emission function for Ha is not very accurately known, much more could be learned of the variations in the angular and velocity distribution of protons if simultaneous observations were made of the zenith and horizon profiles of hydrogen lines emitted from the same part of the aurora. This is because variations in the two distributions affect the relative widths of the two profiles differently; for example, a widening of the angular distribution (decreasing n in (1)) would increase the width of the horizon profile, whereas it would decrease that of the zenith profile. On the other hand, a widening of the velocity distribution (decreasing s in (1)) would increase the width of both profiles. There is therefore an urgent need for such observations. 4.2. The adiabatic equation which governs the movement of the protons in the Earth’s magnetic field reads E sin2 0
-
B
=
E, sin2 0, Bo
’
(3)
where E is the kinetic energy of the protons and B the magnetic field(lv”). This equation is also valid if E changes because of acceleration of the protons in an electric field, unless important potential differences occur over distances comparable to the gyro-radius of the protons (and not directed along B),or unless oscillations in the electric field occur, with frequencies comparable to or higher than the gyro frequency. With the same reservations, acceleration in directions across B cannot occur because of the Hall-effect.
0. E. JOHANSEN
1230
and A. OMHOLT
The relation between the angular distribution of the total particle influx, N(e) = r,@) cos 8, at any point, compared to the reference point indexed 0, is then given by (4) where iV(e) and q(0) have the same significance as in $3.1. This may be shown directly or by Liouville’s theorem (lg8). Suppose for example, there is a reservoir of particles with MaxweIlian distribution in E,, with characteristic temperature T,, from which the protons are accelerated to energy E. The distribution of these particles with pitch angle and initiai energy is given by
(5) if we assume N(0,) to be isotropic.
r(e)
cos
e = iv(e) =
cl
By integration over E, we get EB,COS
B
8
*
i
s I,
E,, COS 6$
exp (--EoW,)
dEo,
(6)
where E,,, is the minimum allowable value of E. as given by (3): E,
=
2sins8,
(7)
(since sin2 0, I; 1) and from (3): cos e, =
1 - ;i
sin2 e
0
l/2 = (1 - E,,JEJ1’?
Substituting x = Eo/E,,, gives
v(e)
cos
e = N(e) =
cl
J-0 B
cos e
sco
exp(--JcEmlkTo)
1
dx
(X2 -
x)1/2
-
(9)
This equation shows that with increasing 8 (increasing E,,J IV(e) falls off faster than cos 8, provided E, is not very small. Moreover, a study of the derivatives of the integral in (9) with respect to i? and (EB,/B) shows that the higher EBo/B is, the more narrowly q(e) is concentrated around 8 = 0. On the other hand, if no acceleration takes place so that E = E,, then from (4) and (6) the angular distribution for a given vafue of E is given by
r(e)
cos
e = iv(e) =
cl
Bo
cos
e N(e,)
z
1 - 4 sina e i
l/2*
00)
1
But since the particles probably originate somewhere out in space where B. < B and N(0,) is nearly isotropic over the range of 0, in question, we have '17(e)~0~e = N(e) = C,COS~.
(11)
In this case the distribution function r(e) would be isotropic, regardless of energy, i.e. n = 0 in (1).
VARIATIONS
IN THE DOPPLER
PROFILE
OF Ha IN AURORAE
1231
By draining from a reservoir of trapped particles or stepwise acceleration effected by bouncing between magnetic mirrors, a wide distribution can be expected; the protons would escape and move into the atmosphere as soon as they satisfy (3) with 0 = 7r/2 and B as the value of the magnetic field just above the atmosphere. All particles with low 8 would be drained out immediately leaving only particles with high 8 when a quasisteady state is reached. In general, we may say that if EBo/B > kT,,, then there will be a pronounced fall off in ~(0) with 8, whereas if EBo/B < kT, q(O) will be nearly isotropic. The fall off will be steeper with increasing EB,,/B. The observational evidence suggests that in some cases r(O) falls off as 0 increasesindeed this may always be the case. If q(O) varies as seems likely, it clearly cannot always be isotropic. If the theory outlined above has any resemblance to reality, then this fall off in l;l(O) implies that EBo/B > kT,, where B, and To are the magnetic field and the temperature of the plasma before the protons are accelerated. The distance into space at which acceleration starts is limited if the temperature of the plasma is not low enough to ensure that this relation is fulfilled with the interplanetary value of the magnetic field for B,,. Another point is that with this model protons with higher final energies E before entry into the atmosphere would show a more narrow angular distribution than those with smaller energies. Inspection of the observed profiles may lend some support to this view. Such a variation would, for example, improve the computed horizon profile shown in Fig. 3, as the center peak (caused by slow protons with small 0) would be more spread out and the wings (caused by fast protons with high 0) would be diminished. Changes in the velocity distribution as indicated in $4.1 could then probably give the necessary correction to the computed zenith profiles. We could certainly make a number of trial and error computations with various model distributions and find some which would give a good fit to any profile. Similarly, we could make theoretical computations assuming various conditions in space. However, this hardly seems worthwhile since observational material is still meagre and unsatisfactory, and uncertainties still exist in the basic emission function for a proton-hydrogen beam in air. Nevertheless, observations of this kind have great potentiality. Acknowledgements-The work reported in this paper has been supported in part by the Cambridge Research Laboratories, OAR, through the European Office, Aerospace Research, U.S. Air Force, contract AF 61(052)252. It is also a pleasure to express our thanks to the staff at the Auroral Observatory for valuable help in various ways, and to Professor D. R. Bates, F.R.S., for helpful comments on the manuscript. REFERENCES 1. J. W. CHAMBERLAIN, Physics of the Aurora and Airglow, p. 264. Academic Press, New York and London (1961). 2. Y. I. GALPERIN,Planet. Space Sci. 10, 187 (1963). 3. R. MONTALBETII,J. Atmos. Terr. Phys. 14,200 (1959). 4. J. M. MALMLLE.,Planet. Space Sci. 2, 130 (1960). 5. A. OMHOLT,W. STOFFREGEN and H. DERBLOM,J. Atmos. Terr. Phys. 24,203 (1962). 6. W. STOFFREGEN and H. DERBLOM,Planet. Space Sci. 9,711 (1962). 7. J. W. CI-L4MBEIUArN, Sky & Telex. 17, 339 (1958). 8. A. OMHOLT,Geofys. Publ. 20, No. 11 (1959). 9. L. H. MEREDITH,L. R. DAVIS, J. P. HEPPNERand 0. BERG, Z.G. Y. Rocket Report Series. No. 1, p. 169 (1958). 10. C. E. McILw~, J. Geophys. Res. 6!5,2727 (1960). 11. H. AL&N, Cosmical Electrodynamics. Clarendon Press, Oxford (1950).