By assuming the validity of a subset of the Rankine-Hugoniot conservation relations for interplanetary (IP) shocks in an isotropic medium it has been demonstrated, in principle, that improved shock normals can be calculated by using a least-squares technique on combined magnetic field and plasma data from a single spacecraft. The scheme uses those six conservation relations not involving pressure and temperature. This paper deals with a test of the scheme by examining in detail a shock across which the magnetic field changed direction by a small amount (approximately 10 deg). On January 26, 1968 at about 1430 UT this shock was observed by the plasma and magnetic field instruments in Explorers 33 and 35. The spacecraft were 76.6 and 56.9 R sub E sunward of the earth, respectively (and 43.5 R sub E from each other), and therefore well outside the earth's bow shock region, a necessary condition for a valid test

Lepping, R. P.

English

Calculating improved shock normals using least squares technique on combined magnetic field and plasma data from single spacecraf

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i NMA T M X,	 41
X-692-71-138
"REPRINT
A TWO. SPACECRAFT TEST OF
A SINGLE SPACECRAFT METHOD
OF ESTIMATING SHOCK NORMALS
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MARCH 19711
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----- 
GODDARD SPACE FLIGHT Cf.NTER
GREENBELT; -MARYLAND
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X-692-71-138
A TWO SPACECRAFT TEST OF A SINGLE
SPACECRAFT METHOD OF ESTIMATING SHOCK NORMALS
by
R. P. Lepping
Laboratory for Extraterrestrial Physics
NASA Goddard Space Flight Center
Greenbelt, Maryland
March 1971
Submitted to the Proceedings of the Solar Wind Conference, Asilomar,
Pacific Grove, California, March 21-26, 1971.
IAbstract
By assuming the validity of a subset of the Rankine-Hugoniot conservation
relations for interplanetary (IP) shocks in an isotropic medium it has been
demonstrated, in principle, that improved shock normals can be calculated
by using a least squares technique on combined magnetic field and plasma data
from a single spacecraft. The scheme devised by Lepping and Argentiero (1971)
uses those six conservation relations not involving pressure and temperature.
This paper deals with a test of the scheme by examining in detail a shock
across which the magnetic field changed direction by a small amount (;ZZ100).
On January 26, 1968 at about 1430 U.T. this shock was observed by the plasma
and magnetic field instruments on Explorers 33 and 35. The spacecraft were
76.6 and 56.9 earth radii (RE) sunward of the earth respectively (and 43.5 RE
from each other), and therefore well outside the earth's bow shock region,
a necessary condition for a valid test. It was assumed that an IP shock's
surface is locally plane over dimensions of about 100 R E . Using this
assumption and the known geometrical configuration of the positions of the
spacecrait with respect to the earth at the times of the shock onset the
orientation of an "observed" normal was ascertained. For comparison least
squares best-estimate normals were then calculated for each spacecraft using
three different time intervals of data in each case: 9, 12, and 18 minutes,
before and after onset. This was repeated using only the magnetic field
data and the conventional coplanarity theorem for further comparison. For
the 18 minute data interval it was shown that the best-estimate normals for
Explorers 33 and 35 agree with each other within less than 3°, and correspond
to the "observed" normal within its angular uncertainty due to the time
uncertainty of the earth's sudden commencement.
E
1
- 2 -
Introduction
The purpose of this paper is to test a single spacecraft method of 	 j
estimating shock normals by cross checking the results of the method applied
independently to the data of two interplanetary spacecraft located about 44
earth radii (RE) from each other. The method, devised by Lepping and
Argentiero (1971), uses a six equation subset of the eight equation Rankine-
Hugoniot conservation relations for interplanetary (IP) shocks in an isotropic
medium, i.e., those equations not involving temperature or pressure. They
showed in principle that improved normals can be calculated by employing a
least squares technique to best fit the combined magnetic field and plasma
data from a single spacecraft to three equations of the six equation subset,
after transformation to an arbitrary frame of reference. The remaining
three equations are used explicitly to obtain the direction of the normal
and, provided the average pre-shock plasma velocity is sufficiently accurate,
the speed of the shock. The reasons for ignoring the equations containing
temperature or pressure are:
'	 1. The proton data for these parameters usually show the poorest
approximation to a step function of all the shock parameters,
2. Use of these parameters would require electron data which is not
always available, and
3. Probably most importantly, use of the energy flux equation, which
does not take into account possible heat flow across the shock
front (Hundhausen and Montgomery, 1971), is of questionable validity.
Observations and Discussion
This paper deals in detail with a shock whose associated magnetic field
changed direction by a small angle (;=100 ) across the shock transition zone.
I
/­". 1
- 3 -
On January 26, 1968 at about 1430 U.T. this shock was observed by Explorers
33 and 35 with an 88.8 + 3.6 sec, time delay between them (see Figure 1).
At about 1441 U.T. 2 sudden commencement was observed on earth. Approximately
two hours later (1634 U.T.) Pioneer 8, located about 570 R E behind the earth
near the earth's tail, saw the shock after some deflection of its normal's
direction. Notice that the IP shock normal was southward by 20 0 but at the
Pioneer location it had become northward by ,=350 (95% certainty error cone
angle is 17 0 ). Only Explorer 33 was significantly out of the ecliptic plate
and was 26 RE below it. The IP shock normal, n, was almost perpendicular
o(,::^70 ) to the preshock magnetic field direction, B 1 . Therefore, little
change of direction of the magnetic field would be expected as the shock
A	 A
passed the two spacecraft. The quantities n and B 1 are best fit values,
whose estimates will be discussed below.
Figure 2 shows superimposed magnetic field data from Explorers 33 and 35
3
around the time of the shock. There was essentially no change in t3 across
the shock surface and only about 10 0 change in 0. The horizontal lines
represent the average of the two individual Explorer 33 and 35 best estimate
values. The length of these lines indicate the 18 minute time intervals,
3
before and after shock onset, that were used in the best	 fit calculation.
All six best fit magnetic field parameters seem to have reasonable values
when compared to straightforward averages allowing for deviations equal to
the rms deviations for each.	 Notice the occurrence of a periodic structure
before and especially after the shock.	 Behind the shock the oscillations,
occurring over about 30 minutes or so, are clearly out of phase between the
Explorer 33 and 35 observations.
Figure 3 shows the plasma data also superimposed from Explorer 33 and 35
observations. The horizontal lines in the pre-shock case are simply averages
of the dual spacecraft data. However, plasma velocity differences are
i"
- 4 -
obtained from the best fit scheme and these along with the added pre-shock
averages yield the post-shock "best-fit" values shown. Again the lines
represent an 18 minute interval before and after the shock. Notice that the
periodic structure after the shock, which was rather clear in the magnetic
field data, also appears here except the wave-like signature is not now quite
as well defined.
Table 1 gives the best estimate values of the IP shock parameters for
the two spacecraft and average values of these best estimates. The subscripts
1
1 and 2 refer to pre- and post-shock, respectively, and the R-T-N coordinate
system, centered at the spacecraft of interest, refers to the unit vectors:
R, radially away from the sun in the ecliptic plane; T, perpendicular to R
AA	 A
and lying in the ecliptic such that R x T = N is normal to the ecliptic and
"northward". W = (W R1 W T , W N ) is the plasma bulk velocity difference V2-V1.
The N's are the number densities and n R , n  and n  are the components of the
shock unit normal. The Alfven mach numbers for pre-and post-shock were 8.5
and 5.5, respectively; these compare well with those of previously studied
IP shocks (Hundhausen, 1970). The best estimate Explorer 33 and 35 normals
(calculated from the 18 minute interval) differed by less than 3°. The
associated calculated shock speeds were 507 km/sec and 520 km/sec respectively,
giving an average value of 513 km/sec. An average pre-shock plasma bulk
velocity V1 = (426, 17.4, -7.0) km/sec from the data of both spacecraft was
used.
i
- 5 -
Parameter
TABLE 1
January 26, 1968 Shock Parameters
Best Estimate Values for the 18 Minute Interval
Best Estimate (B.E.) Value	 Average of B.E. for
Exp. 33	 Exp. 35	 Exps. 33 and 35
iI.
BIR	
(y) -1.59 -0.24 -0.92
BIT -3.16 -3.07 -3.12
B IN -3.49 -3.83 -3.66
B 2
-5.15 -3.41 -4.28
B 2
-6.60 -6.09 -6.35
B 2
-7.54 -8.26 -7.90
w  (km/sec) 78.6 85.6 82.1
WT -37.0 -35.0 -36.0
w 
-28.3 -24.2 -26.3
N I	(#/cm3 ) 4.19 4.45 4.32
N2 9.67 10.52 10.1
n 
0.826 0.850 0.838
-0.440 -0.416 -0.428nT
-.352 -0.324 -0.338nN
Fi r I.
- 6 -
Two Spacecraft Test
The best fit IP normal was checked for accuracy by comparing its angular
displacement from two fixed and intersecting lines in space. These lines
were: first, the segment between Explorers 33 and 35 and, second, that between
33 and the earth; they intersected at 47°. Each of these angles can be
calculated in two ways: first, by a straightforward calculation using the
best estimate normal; this gives the calculated check-angles, and, second,
by assuming,for dimensions of about 100 RE,
(a) a plane shock front. and
(b) a constant shock speed (513 km/sec) and constant normal;
the latter are the observed check-angles. The calculated and observed
check-angles can then be compared.
In the case of the Explorer 33-35 line the observed and calculated angles 	 j
were 80.5° and 84.1° respectively, giving less than a 4° difference. In the
case of the Explorer 33-earth line the observed and calculated angles were
	
, r". 1
47° and 44° respectively, giving approximately a 3 0 difference. The sudden
commencement (SSC) time at earth was taken to be 1441 U.T., giving an 11
minute delay. If 1440 U.T. is taken as the SSC time, giving a 10 minute
delay, the angles become 52° and 44 0 respectively, an Pz60 difference.
Assumptions (a) and (b) for the region from Explorer 33 to the earth, even
over the path from the bow shock encounter to the earth, are justified within
an angle error of 8° or so, because the chock "effectively spends" only
about one tenth of its total 33-earth travel time in this latter region.
The (95% certainty) error cone angle associated with the normal was 7.60
t
which is consistent with the check-angles, or is perhaps somewhat conservative.
r
LLJ
iComparison of Analysis-Intervals
To obtain some understanding of the importance of using the proper time
interval around the shock for the shock analysis, other time intervals, as
well as the 18 minute interval, were used. Heiceforth, the term "best
estimate" refers only to a given analysis-interval for a given spacecraft,
and not necessarily to the final best estimate of the IP normal. Figure 4
shows, for the three separate input data intervals, estimates of the January
IP shock normal, as projected on the R-T plane, for both Explorers 33 and 35.
The results of both the average magnetic field method (coplanarity theorem)
and the best estimate method (auxiliary use of plasma data) of estimating
the IP normal are shown. The latter are represented by either dashed or	 J
solid arrows, and the former by dashed or solid lines; dashed indicate
Explorer 33 estimates and solid indicate Explorer 35. The following features
should be pointed out:	 , ."pl.
a. There is a large (70 0 ) spread of the average normals but a reasonably
narrow (140 ) spread of the best estimate normals over the three time
intervals.
b. Lengthening the time interval of data around the shock for use in
calculating the normal does not necessarily improve the estimate,
even within the short range considered here (i.e. up to 18 minutes).
C. For each given time interval the best estimate normals between
Explorers 33 and 35 are closer together than the average normals.
•	 d. The 18 minute interval was clearly the "proper" choice of interval
•	 giving a few degrees difference between the Explorer 33 and 35 best
estimate normals.
Figure 5 corresponds to Figure 4 except now the estimates of the IP
normal are projected into the R-N plane. All of the above comments again
- 8 -
hold except for statement "c". (However, in statement "a" the spread angle
for the average normals becomes 46° but that for the best estimate spread
remains 14°).
If one had been satisfied with only 12 minutes ("B"-interval) of data
around the shock and had no* taken advantage of the available plasma data
(or did not have such data) one might have been led into a false sense of
certainty about the results because of the relatively good agreement between
the results of the two spacecraft for this time interval.
Conclusions
We have accurately estimated an interplanetary shock normal and have
A
shown its direction to be significantly different from the R-direction both
j	 in inclination angle 9 and azimuthal angle 0 (9= -20°, 0 = 153 0 ); the ecliptic
plane projection was approximately along the average magnetic field spiral
direction. There was no obvious solar flare associated with this shock.
The shock may or may not have originated at the sun but it probably did
not start as a spherical front near the sun unless the front was severely
distorted over 1 A.U. The periodic structure occurring behind the shock as
seen especially in the magnetic field data is no doubt, in part, responsible
for the fact, stated above, that lengthening the analysis interval does not
necessarily improve the estimate of the normal. A proper analysis-interval
is probably one that encompasses, as exactly as possible, two oscillations
if such quasi-periodic structure exists after the shock, or at most should
be limited to the interval just up to the first obvious discontinuity appearing
after the shock.
.
tv
- 9 -
ACKNOWLEDGMENT
I wish to thank Dr. N. F. Ness for the magnetic field data and
Dr. J. Binsack and Mr. H. Howe of M.I.T. for the plasma data used in this
study. I am grateful to my colleagues at GSFC for helpful suggestions
and comments.
„r_... ^.
- 10 -
FIGURE CAPTIONS
Figure 1	 The positions of Explorers 33 and 35 and Pioneer 8, at the
time of the January 26, 1968 shock, shown in the ecliptic
A
R-T plane. Also shown are the best fit IP normal, r., and
A
pre-shock magnetic field direction, B„ as well as a roughly
estimated normal (using average magnetic fields) at Pioneer
8. Quantities in parenthesis refer to the direction
perpendicular to the ecliptic plane in either degrees or
RE. The question mark (?) at Pioneer 8 refers to the large
uncertainty (error cone 17 0 ) of the normal's estimate at
that location.
Figure 2	 Superimposed magnetic field data for Explorers 33 and 35.
F is the magnitude, 0 is the azimuthal angle measured
counterclockwise in the ecliptic plane from 0 = 00 in the
direction of the sun, and 9 is the angle of inclination
measured positive northward from the ecliptic.
Figure 3	 Superimposed plasma data for Explorers 33 and 35. W is the
thermal speed and V is the magnitude of the bulk plasma
velocity, whose direction is designated by 9 (same as in
Figure 1) and 0 (¢ = 00 in antisolar direction). N is
the plasma number density.
Figure 4
	
Estimates of the January 26, 1968 shock normal from Explorers
33 and 35 data and projected into the R-T plane. Both
average-field and best-fit methods are shown, each for three
separate data intervals.
Figure 5
	 Estimates of the January 26, 1968 shock normal from Explorers
33 and 35 data and projected into the R-N plane.
I
/" , I
- 11 -
REFERENCES
Hundhausen, A. J., Composition and dynamics of the solar wind plasma,
Rev. of Geophys. and Space Phys., vol. 8, no. 4, 1970, esp. pp.786-87.
Hundhausen, A. J., and M. D. Montgomery, Heat conduction and non-steady
phenomena in the solar wind, Preprint from Los Alamos Scientific
Laboratory of the University of California, LA-DC-11938, 1971.
Lepping, R. P., and P. D. Argentiero, A single spacecraft method of estimating
shock normals, to be published in the J. Geophys. Res., 1971.
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4 1
JANUARY 26,1968 SHOCK OBSERVATIONS
PLASMA DATA
	
EXPLORER 33 ••••••	 EXPLORER 35
100
w	 50
(KM/SEC)	 '•'••^'^'w =` ,^1.'+ »• ' ^ ^-0
+10°
8 00
• ••	 ••	 •	 •	 ••	 •
-10°
+100
Oo
• •	 J^ S^`n ^yC 	 • •	 • •
-100	 •1
550
500-
V 450
(KM/SEC) 400 ' f^,,,. 	 ^``•	 •
350
50
10- +	^'..^
N	 5-
I
1200 1300
	 1400	 1500
	 1600	 1700 U.T.
Figure 3
0 0	 02	 0.4	 0.6	 0.8	 1.0
T
Figure 4
31-
R
1.0
I
.JANUARY 2F 19AR SHnrK NnRMAIT-
0.
AVERAGE FIELD NORMAL
35
BEST ESTIMATE NORMAL
35
A=9
B=12
	
INTERVAL LENGTH (MINUTES)
C=18
0.2-
N	 0.0
-0.2
A
\	 ^^ B
\	 A
B
B
	
-1.0 1 	 1	 1	 1	 1
0.0	 0.2	 0.4	 0.6	 0.8
Figure 5


A two-spacecraft test of a single spacecraft method of estimating shock normals

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