Analytical tools are presented in two reports. The first is a probability analysis of the orbital distribution of events in relation to dust flux density observed in Pioneer 8 and 9 distributions. A distinction is drawn between asymmetries caused by random fluctuations and systematic variations, by calculating the probability of any particular asymmetry. The second article discusses particle trajectories for a repulsive force field. The force on a particle due to solar radiation pressure is directed along the particle's radius vector, from the sun, and is inversely proportional to its distance from the sun. Equations of motion which describe both solar radiation pressure and gravitational attraction are presented

Wolf, H.

English

NASA Technical Reports Server

(Final Report)
ANALYTICAL TOOLS
AND ISOLATION OF TOF EVENTS
H. Wolf
N74-20175
(N( As-CR--3294 6) ANALYTICAL TOOLS ANDN7 1
7
ISOLATION OF TOF EVENTS Final Report
l(Analytical Mechanics Associates, Inc.)
23 p c S4.25 CSCL 12A UnclasG3/19 34274
AMA Report Nos. 74-5 and 73-45
Contract NAS5-11992
January 1974
ANALYTICAL MECHANICS ASSOCIATES. INC.
9480 LANHAM SEPERN ROAD
S RBROOK. MARYIAND 20801
https://ntrs.nasa.gov/search.jsp?R=19740012062 2020-03-23T11:14:47+00:00Z
SUMMARY
This final report consists of two reports concerning analytical tools.
The first report is entitled, "Probability Analysis of the Orbital Distribution of
Events". It was included as an Appendix in AMA Report No. 73-44 (October 1973).
The second report is entitled, "Particle Trajectories for a Repulsive Force Field".
These two reports summarize the work done on Task III of the subject contract.
The work on Task IV was included as a modification to the program re-
ported in AMA Report No. 73-44 (Statistical Information Program, Progress Re-
port, (Task II)), October 1973.
The modification works fully automatically and points out TOF events. The
modified program has been checked thoroughly and delivered to the Technical
Officer. So far, two TOF events were discovered.
The work on Task III and IV of the subject contract has thus been com-
pleted and will be invoiced.
PROBABILITY ANALYSIS OF THE ORBITAL
DISTRIBUTION OF EVENTS
H. Wolf
AMA Report No. 74-5
Contract NAS5-11992
January 1974
'P"'ING PAGE BLANIK NOT :
ANALYTICAL MECHANICS ASSOCIATES. INC.
10210 GORINBELT ROAD
SEABROOK. MARYLAND 20801
Table of Contents
Page
Introduction ........................... ........................... 1
Analysis ............... .............................. ...... 1
Application to Actual Orbital Distribution ............................ 2
Conc lusions ..................... *........................  4
References ................................ . . .. . . . , .,... 4
iii
PROBABILITY ANALYSIS OF THE ORBITAL
DISTRIBUTION OF EVENTS
1. Introduction
The fine structure of the dust flux density may be examined by noting
asymmetries in the event rate. The analysis contained herein permits a dis-
tinction between asymmetries caused by random fluctuations and systematic
variations by calculating the probability of any particular asymmetry. The
analysis is applied to an asymmetry observed in Pioneer 8 and 9 distributions.
2. Analysis.
Let the relative probability of an event occurring in the half orbit HT
as the spacecraft is moving toward the solar apex be p, and in the other half
orbit HA be q, where p +q=l.
If n events are observed altogether, the probabilities of all possible
combinations are given by the terms of the binomial expansion
n n n \n-1 n n-2 2
+q) =p + 1  q+ p q +...
Thus the first term in the expansion gives the probability of all events occurring
in HT, the second term the probability of all except one of the events occurring
in HT' etc. Other probabilities may be obtained by summing appropriate terms
in the expansion (1). For instance, the probability of at least 7 of the events
occurring in HT (i.e., at most n- 7 events in H ) is given by the sum
T n-y A(n-  n-k k
k=O
It is apparent that for large values of n, the computation of the numerous terms
involved in (2) may be extremely1 ohorious.a However, for nnn > 1 an adequrate
approximation to the binomial distribution is given by:
(n k n-k 1 -(k-*ip)2/2npq
kp - /2?rnpq
(See Reference 1);
1
i.e., the binomial distribution may be approximated by a Gaussian distribution
of mean n p and standard deviation n .
Instead of summing the terms involved, in an expression such as (2), we
may use the corresponding extensively tabulated integrals of the normal curve.
3. Application to Actual Orbital Distribution.
The actual distribution for Pioneer 8 events was found to be:
Total number of events analyzed: 230
Events in HT (moving toward s.a.) 139
Events in HA 91
Corresponding numbers for Pioneer 9
Total number of events analyzed: 111
Events in HT  73
Events in HA 38
Combined totals:
Total number 341
in HT 212
in HA  129
As a working hypothesis it is assumed p = q = .5 (i.e., the probability of
an event occurring is the same for each half orbit).
For Pioneer 8. The expected number in each half orbit is then np = 115
and the deviation from this expected number is 139 - 115 = 24, thus for this case
a = =230 • 1 1  7.58. Consequently, we have a deviation of 3.17 a, corres-2 2
ponding to a probability of 0.16%.
2
For Pioneer 9 the deviation from the mean is 17.5 and the standard deviation
is 5.27 resulting in a deviation of 3.32 o. This corresponds to a probability of
0.08%.
It is realized, of course, that the time spent in each of the half orbits is
not the same. Pioneer 8 spends 578 days in HT and 507 days in HA corres-
ponding to a daily event rate of .240/day and .179/day, respectively. If we in-
crease the time in HA by 71 days and hypothesize that the daily event rate holds
this would indicate 13 additional events occurring in the half orbit if A.
So, the new hypothetical distribution would be:
(Pioneer 8)
Total number of events 243
Events in H 139
Events in HA 104
np=121.5
d = 17.5
S= 7.79
Thus d = 2.24 a, corresponding to a probability of 2.5%.
For Pioneer 9, the event rate in HA was .113 and the number of days
spent in HT and HA was 35.- and 335, respectively.
Making a same adjustment as above the additional 20 days would lead
to an additional 2 events in HA . Thus, the new hypothetical distribution would be:
(Pioneer 9)
Total number of events 113
Events in HT  73
Events in HA 40
Expected number, np = 56.5.
3
With the deviation d = 16.5, and the standard deviation a= 5. 32, then
d = 3.10 a. This corresponds to a probability of 0.2%.
The combined results are tabulated as:
Hypothetical:
Pioneer 8 and 9
Total number of events: 356
Events in HT 212
Events in HA  144
Expected number of events in each half orbit: 178
d = 212 - 178 = 34
a= 9.43
d = 3.610,
corresponding to a probability of 0.04%.
4. Conclusions.
It is seen that the probability for random variations to produce the
observed asymmetry in the distribution of events is very small. Then, the
observed asymmetry should be attributed to some systematic effect, such as
the presence of interstellar grains.
5. References.
[1] "Probability, Random Variables, and Stochastic Processes",
Athanasios Papoulis (Polytechnic Institute of Brooklyn).
McGraw-Hill Series in Systems Science. Copyright 1965.
4
PARTICLE TRAJECTORIES FOR A
REPULSIVE FORCE FIELD
H. Wolf
J.B. Eades, Jr.
AMA Report No. 73-45
Contract NAS5-11992
October 1973
PR EEDING PAGE BLANK NOT fL I 
)
ANALYTICAL MECHANICS ASSOCIATES. INC.
10210 OREENBZLT ROAD
SEABROOK. MARYLAND 20801
Table of Contents
Page
Introduction ...................................................... 1
Computational Sequence ......................................... 3
Typical Results ........................................ ...... . 6
Reference ...................................................... 13
iii
Introduction
Under simple assumptions the force on a particle due to solar radiation
pressure is directed along the particle's radius vector, from the sun, and is in-
versely proportional to its distance from the sun. In addition the force is assumed
proportional to the cross sectional area of the particle. Thus the particle's
acceleration due to radiation pressure is given by:
CeA R
m 3
r
and the equation of motion of the particle subject to both solar radiation pressure
and gravitational attraction is given by:
R CA R R
m 3 - 0(1-8) 3 ()
r r r
where:
R is the radius vector of the particle
r its magnitude (= IR )
,u, the sun's gravitational constant (= K2M )
CO the force due to solar radiation pressure on a particle
of unit cross sectional area at unit distance
A the cross sectional area of the particle
m particle mass
GA
= (2)
AO m
If the particle is assumed approximately spherical then
A Tra 2  3
m 4 3 4ap
-wa p3
where
a is the particle radius and
p its density
thus 3C 1 k
Gi (4)4/( pa (mp2)1/3
defining the constant k as:
1
=97 1/3
16/ g
For large, dense particles << 1; consequently the normal Keplerian
orbit for such a particle would only be slightly modified. As the particle radius
or density decrease 0 will increase and may, for sufficiently small, light
particles, exceed 1. In this case the net force will be repulsive.
Kepler's theory may be modified for these cases. The motion then takes
place along the branch of the hyperbola which is convex toward the focus occupied
by the sun.
For situations such as these there will be a "forbidden zone" about the
sun into which these particles may not penetrate. The zone is of course de-
pendent on the "velocity at infinity", the radiation pressure and the angular
momentum of the particle. It is the purpose of this note to develop a sketch
of this "forbidden zone".
The value of the radiation pressure constant used here, has been chosen
to be consistent with that given by Silverberg. (Reference [1]).
The attractive and repulsive cases are separated by = 1; since here the
particle moves in a force free field. For a particle of given mass this gives rise
to a "limiting" density; or, for a particle of given density, to a 'limiting" mass or
diameter. Particles with densities (or masses) smaller than their limiting values
are repelled, others are attracted.
-13
As an example: for a particle of mass 8.9 x 10 grams ( similar to that
for event 20 from Pioneer 8) the limiting density of 0. 953 gr/cc.
For a particle with a (very likely) density (p) of 3 gr/cc, its limiting dia-
-5
meter would be 3.85 x 10 - 5 cm, and it would have a corresponding limiting mass
-14
of 8. 99 x 10 gr.
For the curves plotted and presented herein, a density of 3 gr/cc and an ex-
cess speed (V,) of 20 km/sec (corresponding to the velocity of the solar system)
2
has been assumed. (Of course, corresponding curves for other values of these
parameters can readily be obtained).
A word description of the trajectory 'flown" by a test particle in the presence
of the assumed solar mass-radiation influence is given below. In addition, there
is a sketch included here to show several geometric properties of the trajectory.
The flight paths of interest are those which describe hyperbolae in the
vicinity of the sun. These particles are assumed to "arrive" in the solar sys-
tem with an excess speed (V) of 20 km/sec (the assumed value). If these
particles approach the sun not along a direct radius, but at some offset distance
(i.e., with non-zero angular momentum), then they will describe hyperbolic arcs
which can be predicted from the two-body model described above. On the sketch,
below, '1" is the closest solar approach distance; c is the off-set distance
(measured from the solar apex radial), and VC is the approach excess velocity.
Also, as shown on the figure, 8 is an angle locating the hyperbola's axis
measured from the solar apex radial. Due to symmetry, these hyperbolae may
be oriented symmetrically about the radial. Thus there could be mirrored
images of each path on this principal radius.
Computational Sequence
In determining geometric characteristics of these trajectories, and ob-
taining a description of the FORBIDDEN ZONE, the following calculations pro-
cedure has been utilized.
(1). A density of p = 3 gr/cc is assumed; knowing mll m (the limit
mass), choose values for the particle mass (m < mlim)'
(2). With k known, (9.32 * 10 - 5 ) and having chosen p then calculate
a 0 corresponding to each m; i.e., from (4)
-5(9.32* 10 )3.0).
2 M)1/3 (p 3.0).(p3m)
3
/-
Sketch depicting a particle (P) moving from n, with a velocity V, offset at a
distance c, which has a closest approach (q) to the sun (p), as shown.
This closest approach depends on the quantities A, c, V,.
The path for P is a hyperbola (for this repulsive system); the axis of the hyperbola
is Inclined at an angle from th radial (to C). Here, A is denPndnt on the nPrn-Li LU ALLAVU eLL CUA CWIZ U a dial
meters p, c, V,.
4
Note: Here, 6 1. 0, by definition.
(3). Having obtained B, calculate the equivalent A. (sun value) for the
system:
-4 3-
with 1 =2.959 10 - 4 (AU3/DA2).
(4). Next, assuming V. = 20 km/sec, calculate the parameter, B, where:
B -/c
V2
And, (5). Knowing B define characteristics for the hyperbolic path; e.g.:
2
H tan ( tan
and
_q -1i /c + ( - /c 2 ]1/2
c V 2  VC
=B+ 1i +B2]1/2
These data may be used to plot a boundary defining closest approaches to the sun,
for various system constraints and/or conditions. However, the FORBIDDEN
ZONE, as defined, is not yet obtained. This region of "space" lies on the sun's
side of the geometric envelope for the family of hyperbolae traced out by the
assumed particles. In this regard, then, it is necessary to examine the pro-
blem (mathematically or graphically) and to define the envelope accordingly.
Since the model used for the present investigation leads to analytical re-
sults, the envelope curve can be described mathematcally. For this description
1
the parameter (A) is introduced, wherein A E (see above), and the equations
for the trajectory are rewritten. Making use of the formal operations to deter-
mine an envelope, it can be shown that the pair of parametric cartesian equations
for this curve are:
5
x =(A )21-* 1+A]- y+2 (Ac)[1+ 1A
2 ]
(Ay)[J 1+A -2]
and
and (2 + 3A 2 )[1 + 1A 2 ]+A 2
[1+ A23/2
For these expressions the x-axis parallels the solar-apex radial; and its positive
direction is in the apex direction. The y-axis is normal to x; it may be assigned
signs of ± arbitrarily. (Recall that c is the off-set distance -- for V -- ex-
pressed as a distance, measured in AU. Consequently, the coordinates (x, y) are
similarily dimensioned).
Typical Results
Table I shows: (1), a summary for the calculations outlined above. Here
one will find tabulated, as a function of m, the corresponding values for P and
'(see Eqs. (2) and (3)).
Also, in the table, is a listing of closest approach distance (qmin) for
each of the m values selected. The quantity qmin (in AU) denotes the closest
distance that a particle of mass m can approach the sun under the conditions
assumed. Necessarily, these distances correspond to an offset distance (c) of
zero.
Next, for a unit offset distance (i.e., c = 1.0 AU), typical values of
q(AU) and 0 (deg) are tabulated. These data are illustrative of the results which
may be expected from the calculations procedure outlined earlier.
Lastly, there is a listing of c(AU) values which are required to attain a
q = 2.0 AU (for each value of m defined). Since q = 2 AU has been arbitrarily
chosen as an upper bound of interest, here, then these data define a practical
6
upper limit for the present investigation. (Note: when the notation c < 0 is used
herein it infers that these particles cannot have a qmin as small as 2.0 (AU), for
the conditions implied).
Figure I is a graph of qmin and c, as functions of m, for the information
described above.
Figure II is a polar plot of (q (AU), 6o) for values of 1. (or m) based on
a fixed off-set distance (c) of 1 AU. Since the curves are symmetric about 0=0,
then c may be the off-set on either side of the radial (0 ); this is a typical result
for any value of c (within the range of possible q distances allowed).
Figure III shows q, 0 boundaries for several of the values of m used
here. That is, each boundary corresponds to a chosen value of m (! mlim).
This describes a limit in q for the allowable range in c (AU), at a fixed AL. It
should be remembered that q = 2.0 AU represents the arbitrary upper limit for
this quantity. Note that when m < 30 * 10 - 1 5 gr then q > 2.0 AU; i.e., the
particle is subjected to a large enough radiation (back-pressure) so that it does
not. have sufficient kinetic energy to move in as close as 2.0 AU from the sun,
even for a direct radial approach. (The direct solar approach describes the
qmin noted earlier). It should be apparent that if V0 > 20 km/sec then the
corresponding q-values should be smaller; the particles would be able to move
closer to the sun against radiation pressure.
Figure IV graphically depicts the FORBIDDEN ZONE for this problem's
conditions and constraints. Each curve, sketched on the figure, corresponds
to a selected value of m (< mlim). These arcs describe the geometric envelopes
mentioned earlier; they separate regions, in the vicinity of the sun, into which
the particles may move from those in which they cannot (the FORBIDDEN ZONE).
Due to the obvious symmetry of this situation these arcs are mirrored in the
solar (apex) radial line.
7
TABLE I. Calculated Data
Representative Data
Assume Compute Compute Closest Approach* (Based on c s 1AU) c(AU)
m(gr) 1 M (AU, DA) qmin (AU) q (AU) o (For q = 2AU)
89.9508 * 1015 1.0 0 1.0 900 2.0
85 * 10-15  1.01905 -5.63685 * 10-6 0.0845 1.043 87.580 1.956
80 10 - 15 1.03985 -1. 17924* 10 - 5  0.1768 1.0923 84.950 1.91
75 * 1015 1.06247 -1. 84834 * 10 - 5  0.2771 1.1481 82.110 1.86
70 * 10-15 1.087182 -2.57973 * 10-5  0.3867 1.2119 79.050 1.80
65 * 10 - 15 1.114373 -3.38430 * 10-5  0.5074 1.2854 75.760 1.73
60 * 10 - 15 1.144506 -4.27593 * 10 - 5  0.641 1.3706 72.230 1.648
55 * 10 - 15 1.178187 -5. 2755 * 10-5  0.7909 1.4708 68.420 1.55
50 * 10 - 1 5  1.216219 -6.39792 * 10 - 5  0.9592 1.5887 64.380 1.448
45 * 10-15  1.259692 -7.68428 * 10-5  1.152 1.730 60.060 1.304
40 * 10 - 1 5  1.310132 -9.17681 * 10 - 5  1.376 1. 9017 55.480 1.12
35 *10 - 15 1.36976 -1.094132 * 10 - 4  1.640 2.1135 50.640 0.85
30 * 10-15  1.44199 -1. 30784 * 10-4  1. 9607 2. 3808 45.570 0.294
25 * 10 - 15 1.53234 -1.57519 * 10 - 4  2.3615 2.7281 40.260 <0
20 * 10 - 1 5  1.65066 -1. 92531 * 10 - 4  2.8865 3.2 34.720 <0
*Closest Approach would correspond to c = 0(AU).
V.2o 2 <c
/3 O'/CC
(,4u)
2-4
1
4
,zoyir"e"  _;. 4 o 5 60 AO
)141 (&4)
Figure I. Closest approach (qmin) (on a radial) and offset distance (c), for
q - 2 AU, as a function of mass (gr).
9
070
"o 60
//  70
40 /
x 20 0 Jo 20 3o
Figure II. Description of closest solar approach (q, 8) as a function of particle
mass (m) - or equivalently (0 or .) - for a fixed offset distance (c) =
1.0 AU.
10
Figure III. A plot of q and 6 for various values of c (c not shown). Note: each
curve corresponds to a value of m, as specified.
11o
* -
y (AU
Figure IV. Graphic description of the 'TORBIDDEN ZONE" for a specified
case study.
12
Reference
[1] Silverberg, Eric C., "Interplanetary Dust Streams: Observation by
Satellites and Lidar", Thesis: University of Maryland, 1970.
13


Analytical tools and isolation of TOF events

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