Geometrical aspects of the performance of cosmic ray detector telescopes in non-isotropic particle distributions

NUCLEAR INSTRUMENTS

AND METHODS

IO 4 ( I 9 7 2 ) 4 9 3 - 5 0 4 ;

© NORTH-HOLLAND

PUBLISHING

CO.

G E O M E T R I C A L A S P E C T S OF T H E P E R F O R M A N C E OF C O S M I C RAY D E T E C T O R TELESCOPES IN NON-ISOTROPIC PARTICLE DISTRIBUTIONS T . R . S A N D E R S O N and D . E . P A G E

Space Science Department (ESLAB), European Space Research and Technology Centre, Noordwijk, The Netherlands Received 26 May 1972 The performance of a cosmic ray detector telescope made from two circular discs is investigated. Results of calculations are presented on the following: 1) the geometric factor, 2) the angular response to a parallel beam of particles, and 3) the behaviour in a non-isotropic particle distribution.

The performance in a particle distribution takes into account the effective geometric factor, the path length distribution, and the effect of rotating the detector, with particular application to spinning rockets and satellites. Only rectilinear particle trajectories are considered.

1. Introduction

each approach has certain advantages, we have considered both the analytical and a graphical means for determining the geometric factor. Using the ' shadow method' 1), we first of all determined the angular response of the detector telescope to a beam of particles, such as is encountered at an accelerator, or from a point source. The ' shadow' was

Following earlier work 1) on the geometric factor, a more comprehensive study has been made of detector telescope performance in various particle distributions. The geometric factor is here defined as the ratio of the counting rate to incident flux, when the detector is exposed to an isotropic distribution of particles. Since 3.O

t

,

1'0-'

2

3

4

5

6 7 e s ) 10 °

2

3

4

5

6 7 s 910

R1 -.-.im~ -gFig. 1. Graph of the geometrical factor of a telescope with elements of radius R1 and R2, separated by S. G/R~ is plotted against Rt/S for different values of R~/S.

493

494

T. R. SANDERSON AND D. E. PAGE

then used to determine the behaviour in a distribution of particles. If the telescope is exposed to a flux of the form lof(O), the "geometric factor" as defined in ref. 1 is given by the ratio of the counting rate to I o and must include a reference to the particle distribution. We now propose instead a correction factor, which when multiplied by the geometric factor as defined in the first paragraph, gives the ratio of the counting rate to I o. In this way we retain the true geometric factor, which is a function only of the geometry of the telescope, and introduce an additional quantity which describes that particular telescope's performance in a given particle distribution. Previous work 1) has shown that the contribution to the counting rate comes mainly from some angle of incidence away from the axis of the telescope. One consequence of this is that the average path through which the particles pass is different from that expected on the basis of a normally incident flux. Curves are presented to show how this effect depends upon the telescope configuration. Another important consequence of the telescope characteristics is the varying count rate recorded when the telescope is rotated in a non-isotropic distribution. To find what rates are expected, we have calculated the counting rates for various telescopes when rotated under different distributions. When an unknown distribution is being measured, the experimentally observed variation of counting rate with rotation angle can be compared to a set of such calculations, in order to determine the real particle distribution. Finally we have applied the above results to telescopes placed upon a spinning rocket or satellite, to see how the counting rate varies with the spin angle of the rocket or satellite. Only a few figures illustrating the results can be presented here but plans are in hand to publish a more comprehensive set of curves.

flux, the counting rate will depend upon the pointing direction of the telescope and the type of distribution, so that a definition of the ratio of the counting rate to particle intensity must include some indication of the type of particle distribution. The geometric factor for a detector telescope under an isotropic distribution of particles is derived in refs. 1 and 2. For a telescope made up of two circular discs, of radii R 1 and R2, and separated by a distance S, the geometric factor is:

G = ½~2{(R~+R2+S2)- [(R2 + R22 + S2) 2 - (2 R 1 R 2)2] .} }.

(2)

This is presented in graphical form in fig. 1 where we have plotted G/R~ as a function of R1/S, for different values of R2/S. Here we have restricted the range of the parameters from 10-1 to 10 +1, since this covers nearly all the practical arrangements used for cosmic ray detector telescopes. fO RI

R2

(a)

(b)

R1

R1

2. Geometric factors

The geometric factor of a detector telescope exposed to an isotropic flux of particles of intensity I o particles per c m 2 per steradian per second is given by

G = N/Io cm 2 sr,

(¢)

(d)

Fig. 2. Diagram showing overlap of the top disc on the bottom disc, giving rise to the shadow.

(1) (a) 0=0;

where N is the coincidence rate of the system in counts per second due to particles incident from above. Since the flux is isotropic, the geometric factor defined above is independent of the position of the telescope in the radiation. If the telescope is exposed to a non-isotropic

(b) 0
,

THE PERFORMANCE

OF C O S M I C R A Y D E T E C T O R

two discs. If the telescope is inclined at an angle 0 to the beam (fig. 2b) the counting rate will be reduced by a factor cos 0, since the projected area normal to the beam is reduced by this factor. As 0 is increased, a condition is reached where the particles passing through the first disc do not all pass through the second (fig. 2c). In this case the response falls off quicker than cos 0. Eventually a condition is reached where none pass through both discs (fig. 2d).

3. Angular reslmme Since detector telescope characteristics are frequently measured using a particle accelerator, it is necessary to know the response of a telescope to a parallel beam of particles. In fig. 2 are shown different positions of the telescope with respect to a beam of particles. Only those particles which pass through both of the discs will be recorded by the telescope. In fig. 2a is shown the condition when the axis of the telescope is parallel to the beam direction. In this case, the counting rate is the product of the flux and the area of the smaller of the RELATIVE

COUNTING

495

TELESCOPES

Following the method outline in a previous paper1), we calculated the 'shadow area' of the first disc as seen

RATE

0.9-

s:

0.8-

RI == R2,~R 0.7-

0.4

0.3.

\02

\o,

ko6 o8\,o

0.2-

0.1-

ol 0

I

10

,k 20

i

30

\

i

40

\

i

\\

50

l

60

\ \ \ \ I

I

70

80 e

Fig. 3. Angular efficiency of a telescope exposed to a parallel beam of particles vs 0 (0 is the angle between the telescope axis and the

beam direction) for R1 = R~ = R.

496

T. R. S A N D E R S O N A N D D. E. P A G E

by the second disc (see also appendix). The angular efficiency is then given by: (0) = A,(0) cos 0 , A,(0)

(3)

where t/(0) is the ratio of counting rate at angle 0 to the counting rate at 0 = 0 for a parallel beam of panicles, A,(O) is the shadow area at 0, and A,(0) the shadow area at 0 = 0. We have calculated the angular efficiency for a variety of detector telescope shapes. The results of some of these calculations are presented in figs. 3 and 4. Fig. 3 shows the response for a telescope with discs of RELATIVE

equal diameter. Fig. 4 shows an example of the response of a telescope with discs of different diameters. It can be seen that equal diameter discs give a response which is roughly triangular, whilst discs of different diameter give a response which is roughly rectangular.

4. Response to particle distrihntions When a cosmic ray detector telescope is flown on a satellite, rocket or balloon or when used to monitor a flux on the ground, the telescope response depends upon all the incident particles which are within its acceptance angle. Consequently, it is important to consider the telescope in terms of its response to a

COUNTING RATE

T

0.6-

0.7R1 . - ~ : . = 0.2 S

0.6-

0.5-

0.4-

0.3-

0.2

\o, \o°\o.\,o

0.1

0

,0

20

3~0

4'0

so

I, ol ol oloo 60

8;

7'0 e

Fig. 4. Angular efficiencyof a telescope exposed to a parallel beam of particles vs/9 for R1 # R2.

497

THE P E R F O R M A N C E OF COSMIC R A Y DETECTOR TELESCOPES

So,

distribution of particles, rather than particles at a specific angle of incidence. The counting rate of a detector telescope exposed to a distribution of the form I = Io cos 2 0 has been derived analytically2). So that we could derive the counting rates for any distribution, we employed a numerical analysis method rather than attempting to find exact solution. For a general distribution, I(0), the contribution to the counting rate of a detector element, of area 6A, and from an angle 0, and within a solid angle dO is given by:

~SN = I(0) cos0 6A rf2.

dN = I(0) As(O) c o s 0 d[2. Since dfl = sin 0 d0 dO, integration over 0 and gives the total counting rate, hence:

f2. f./2 N =

I

1.0

do

do

= 2~

I

I(O) As(O ) cos 0 sin 0 dO d ~

(6)

I (0) A s (0) cos 0 sin 0 d0,

(7)

~/2

do

if I(0) is independent of 4 . We have integrated this last expression for a variety of detector telescope shapes, using 1° steps, and for an isotropic distribution the results agreed with the expression quoted in section 2 to better than 1%.

(4)

At any one angle, integration of 6A over the whole detector gives the shadow area, As(O ) (see appendix). CORRECTION FACTOR, F

(5)

R2 S 0

:

.

)

1

~ ~

CORRECTIONFACTOR FOR TO COS e AND Z0 sin e

1.0

•,,T

1.5

.8

cos e

2.0

10 4.0

20

1.5 0.5

sin e

10

.4

0,7

o....~5

.3

.2

.1

0.1

'

'

.

.

.

.

I

1.0

.

.

.

.

R1 S

.

.

'

l 0.0 '

Fig. 5. Correction factors for I = 10 sin 0 (lower half of the figure), I0 cos 0 (upper half of the figure) ;with parameters R2/S and R1/S.

498

T.

R.

SANDERSON

AND

Rather than plot a new set of 'geometric factors' appropriate to each distribution, we have plotted a quantity which we call 'correction factor'. Since the geometric factor as defined here is a measure only of the size and shape of a telescope, it cannot be a function of the distribution the detector is measuring. So that we can describe the behaviour of a telescope with a given geometric factor in a distribution which is not isotropic, we have introduced a correction factor, appropriate to each distribution and each telescope. For a telescope looking along the 0 = 0 direction in a distribution I = Io(0), the counting rate is:

PAGE

F = N/Ni,o.

(1o)

The factor F is a measure of the behaviour of a telescope in a particle distribution, relative to its behaviour in an isotropic distribution. F, of course, is itself a function of the distribution as well as the shape and size of the telescope. In figs. 5, 6 and 7 we have plotted F for various particle distributions as a function of the same parameters used in fig. 1.

5. Path length distribution When energy loss measurements are made of a particle passing through a detector, the amount of energy deposited depends upon both the energy loss per unit path length, and the path length of the particle

where F is our correction factor. For an isotropic distribution, Niso = GIo,

E.

so that

(8)

N = F G I o,

D.

(9)

CORRECTION FACTORF, 0.!

R~/

O

o.g

].~

~

~R 2

0.8 1.2

,,,

(17 1.5 0.6

2.0 3.0

2.0 --

0.4

1.5 0.3

~

1.2 1.0

0.2-Q7

0.5 0.1

0

F 01

i

f

T

,

i

r

,

I

'

1.0

'

'

'

'

' 10.0

R1

Fig. 6. Correction factors for I = I0 sin9 0 (lower half of the figure), -10cos2 0 (upper half of the figure).

499

THE P E R F O R M A N C E OF C O S M I C R A Y D E T E C T O R T E L E S C O P E S

considerably greater than at normal incidence. This of course is a direct consequence of the fact that most particles pass through the detector at some angle between 0 ° and the semi-cone angle of the telescope1). Consequently a calibration made at normal incidence will not be valid for a distribution of particles. Furthermore, due to the distribution of path lengths the energy resolution will be considerably worse than that obtainable with normally incident particles. From fig. 8, it can be seen that this effect is quite pronounced for telescopes with large opening angles, and minimal for those with very small opening angles,

through the detector element. The energy loss per unit path length is not concerned with the geometry of the detector, and so will not be considered here. In the case of the path length through the detector, this is a relatively simple matter for normally incident particles (e.g. when the detector is being calibrated using a collimated particle source). The situation is quite different when the detector is exposed to a distribution of particles. To demonstrate this, we show in fig. 8 the relative number of particles which pass through a path length P in the detector, plotted against P (where P is measured in units of the path length at normal incidence) for a detector telescope exposed to an isotropic distribution of particles. Six curves are shown, for different detector telescope configurations, representing different opening angles. For a detector telescope with a large opening angle, the majority of the particles pass through the detector with path lengths

I-

CORRECTION

FACTOR,

6. Rotation o f a detector under a distribution o f particles

As cosmic ray detector telescopes are frequently used to measure anisotropies, it is important to understand how different telescope configurations affect

F

,.o

R1

o.g 0.5 0.8

CORRECTION

0.7

locos30

FACTOR AND

FOR

lesl'n30

0.7

lo 0.6

3

\\

¢os3 0

12

1.5 0.5 t

3.02"0

0.4 1

4D 3.0

0.3

2.0

--t

1.5 Q2

1.2 1.0

0.1

sin30

0.7 0.5 L ] 0.1

o.1

~ -,

,

,

,

i

,

,

I 1.0

,

,

,

,

,

,

I I 1Q.O

5 Fig. 7. Correction factors for I = Io sin80 (lower half of the figure), Io cos319 (upper half of the figure).

500

T. R. S A N D E R S O N A N D D. E. P A G E RELATIVE CONTRII~UTION TO COUNTING RATE

t' 1 8-

R1

6-

f

5-

RI=R2=R

4-- /

~

_t

I/ 2

2.0

1.0

2.0

2.5

PATH LENGTH

Fig. 8. G r a p h o f relative contribution to the counting rate vs path length through the detector, for various values o f the separation S o f the detector elements (with R1 = R2).

B

A I

I

I

.

}

~---eYml "~.. TYm+l

[

I

X Fig. 9. Axes used in calculation o f respons¢ when the distribution is rotated about the detector. OX, OY, O Z are fixed w.r.t, the detector, O A is the reference direction o f the distribution, and OB is an arbitrary vector.

Fig. 10. Residual area encountered during integration o f a function containing cos" 0' when n is odd.

THE P E R F O R M A N C E OF COSMIC RAY D E T E C T O R TELESCOPES

501

RELATIVE COUNTING RATE, R~

I'

"

Nn 0

$i.2e

0°

91o°

180°

Fig. I I. Variation of counting rate of a telescope with R; = R2 = S = 1, when rotated under various particles distributions, as a function of 0c, the angle between the reference direction of the distribution and the telescope axis.

Z

Fig. 12. Diagrams showing angles used in the integration Of the telescope response. O A is the reference direction of the distribution which is inclined at an angle ~ to the telescope axis, OZ. The shaded surface represents incidence directions for particles which are coming from the lower hemisphere of the distribution.

502

T. R. S A N D E R S O N

A N D D. E. P A G E

such a measurement. Thus, we have calculated the counting rates of different telescopes when rotated under particular distributions of particles. For the purpose of the analysis, we have considered the telescope coordinate system fixed, and rotated the distribution about the detector. Referring to fig. 9 the vector OA (for convenience located in x-z plane) is the reference direction of the distribution, inclined at an angle ~ to the telescope axis. The distribution is of the form lof(O'), with symmetry about the vector OA (where 0' = 0). The vector OB is an arbitrary vector, at position (0, 4 ) with respect to the telescope axis. The particle intensity at OB is thus Io f(O, ¢~). I f the direction cosines are defined as: l = sin 0 cos 4 ; m = sin 0 sin • ; ~ / = cos 0, then the angle between vectors OA and OB is: 0' = c o s - 1(1Ale + m, m b + na nO,

(11)

cos 0' = cos 0cos ~ + s i n 0sin cecos 4.

(12)

ROCKET AXIS MAGNETI

I /

/

/

DETECTOR AXIS

and so:

/

With this relation, the counting rate of a telescope exposed to a distribution of the form Iof(O' ) centred upon OA can be represented by:

N =

As(O)Iof(O, 4) cos 0 sin OdOd~,

j

p

,,EL°

"Y = DETECTOR ANGLE

(13)

j o do where As(8) is the shadow area as defined in section 3.

Fig. 13. Axes used for a telescope m o u n t e d on a spinning rocket.

RELATIVE COUNTING RATE ( R ~ )

FiELD(R1 ANGLE r -

o"

1.0

9d

O

o

~)

3 DETECTOR

o

ANGLE,~(= 10°

Fig. 14. Variation in c o u n t i n g rate o f a telescope o f o p e n i n g angle 20 ° (with R1 = R2) vs spin angle, for a telescope m o u n t e d at 7 = 10° from the spin axis, for different values o f t h e angle between t h e reference direction o f a I = I0 cos 2 0 distribution, a n d t h e rocket axis. It is a s s u m e d that the s y m m e t r y axis o f the I -- I0 cos 2 0 distribution is along the m a g n e t i c field direction.

THE P E R F O R M A N C E OF C O S M I C R A Y D E T E C T O R T E L E S C O P E S

in the upper hemisphere. Referring to fig. 12 it can be seen that at large values of 0, there is a possibility that the integration may include some contribution from the lower hemisphere. Thus, if 0 + ~ > ½ n , the integration over • must be stopped at an angle ~ ' , where ~ ' is given by

We have integrated this expression for various functions f(O'). The integrations were performed numerically using 1° steps for 0 and ¢i. The function f(0,¢~) was evaluated for each 1° step of ~b and integrated using Simpson's rule. In the case of functions which included odd powers of cos 0', the modulus of the function was used, to ensure that the particle flux was always positive. A function such as cos n 0', where n is odd, is shown in fig. 10, at the point where it becomes negative. Also shown is the modulus of the function. If the integration of the modulus is performed a small error will be present due to the extra area ABC included in the integration. To a first approximation this area is given by: A = Y,,,Yr,,+I/(Y,,,+ Y,,,+I),

503

• ' = ½ n + s i n - l ( c o t 0 cot e).

Each curve in fig. 11 is normalized at the maximum value. Hence, the counting rate at a given angle of inclination is given by: (16)

C~ = G I o F R ~,

where G is the geometric factor, is the correction factor for the detector telescope F and the distribution, I0 is the intensity of the distribution at 0' = 0, is the relative counting rate at ~.

(14)

and must be subtracted from the integral as derived by Simpson's rule. After the integration over ~, the function is integrated over 0, using also the values of the shadow function which are appropriate to that telescope. The results of these integrations for a telescope with R1 = R 2 = S are presented in fig. 11 where we show the change in counting rate as a function of ~, the angle of inclination of the telescope to the distribution. As well as functions such as sin n 0', [cos~ 0'], we show (1 +cos 0'), and also isotropie and cos ~ 0' distributions which exist only in the upper hemisphere. The counting rates for the last two types of distributions are obtained in a similar way to the others, except that contributions can only be made to the counting rate from particles

7. Application to rocket and satellite measurements The diagram presented in fig. I 1 shows the counting rate of a detector telescope as a function of ~, the angle of telescope inclination to the reference direction of the particle distribution. From a set of such diagrams, it would seem that the most suitable detector telescope configuration for measuring anisotropies is one which has a small opening angle, since that gives the largest ratio of counting rate between maximum and minimum. However, it must be remembered that such a FIELD ANGLE (~)

RELATIVE COUNTING RATE (R~) 1.O

o"

P

(15)

=_

~

aSo° DETECTOR ANGLE, ~t = 80°

Fig. 15. A s fig. 14, b u t telescope m o u n t e d at ~ = 80 ° f r o m tho spin axis.

504

T.R.

S A N D E R S O N A N D D. E. P A G E

particle detector has a much smaller counting rate than one with a large opening angle for the same detector area, so that a balance must be found between a high counting rate and a suitable opening angle. An example of a detector telescope mounted upon a spinning rocket is shown in fig. 13, where the detector is inclined at an angle 7 to the rocket axis. As the rocket spins, the detector sweeps through pitch angles from ( f l - 7) to (fl + 7), where fl is the angle of inclination of the rocket to the magnetic field. In figs. 14 and 15 we have plotted the counting rate variations of a detector mounted upon a spinning rocket, and exposed to a distribution of I = I o cos 2 0. The curves are normalized to unity at g~ = 0 [~ = 0 corresponds to the detector pointing along a pitch angle (fl-7)]. The diagrams are for a detector with two equal discs, spaced apart to give a 20 ° opening angle. For example, if the rocket is aligned at 60 ° to the magnetic field, then the detector mounted at 80° gives completely different results from the same detector mounted at 10° from the rocket axis. In one case, it gives a single peak, and in the other a double peak. This sort of behaviour is extremely important in measuring anisotropies and angular distributions. The actual counting rate of a detector at spin angle g~ is given by C = C~p_r) R~,

(17)

where Ctp_ r) is the counting rate at angle of inclination ( f l - 7) (from section 6) and R~ is the relative counting rate at spin angle • (from diagrams 14 and 15).

first detector element upon the second in the direction of the angle of incidence under consideration. In figs. 2b-2d are shown three different angles of incidence, representing the following three regions. (a) Fig. 2b: 0 < tan-~

[R2-Rl___.7[.

In this case, As = rcR22, if R 1 > A s = r c R 2,

R2,

i f R 2 > R 1.

(b) Fig. 2c: tan- 1 I -R-27--R- 1~

< 0 < tan-I (~A-~)"

In this intermediate range, the shadow is given by:

As(O) =

R ~ ( X - s i n Xcos

X)+R22(r-sin

Ycos Y),

where X = cos-1 [(R 2 _ R 2 + D2)/2 DRI)], Y = cos-I [(R 2 _ R~ + D2)/2 DR2], and D = S tan 0. (c) Fig. 2d: 0 > tan

The authors are indebted to Dr. E. A. Trendelenburg for making this work possible.

Appendix

Calculation of the shadow As(O). The 'shadow' term As(O) referred to in section 3 is the projection of the

Here,

As(O) =

0, since there is no overlap.

References 1) v. Manno, D. E. Page and M. L Shaw, ELDO/ESRO Sci. Tech. Rev. 2 (1970) 363. 2) D.J. Heristchi, Nucl. hlstr, and Meth. 47 (1967) 39.