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ATTITUDE DETERMINATION OF SPIN STABILIZED SPACECRAFT

USING A DIGITAL ASPECT SENSOR

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MARCH 1965

GODDARD SPACE FLIGHT CENTER GREENBELT, MARYLAND -

https://ntrs.nasa.gov/search.jsp?R=19650012054 2018-06-14T17:38:05+00:00Z

X-631-65-13

ATTITUDE DETERMINATION

OF SPIN STABILIZED SPACECRAFT

USING A DIGITAL ASPECT SENSOR

March 1965

. Goddard Space Flight Center

Greenbelt, Maryland I

.

LIST O F ILLUSTRATIONS

Figure Page

1 Euler rotation angles . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Relationship of energy to precession half angle for constant momentum ................................. 6

3 Celestial sphere representation of a precessing satellite using a digital aspect sensor . . . . . . . . . . . . . . . . . . . . . 7

4 Quantized field of view of a digital aspect sensor . . . . . . . 7

5 Typical plots of sun sightings from digital aspect sensors. . 9

6 Plot of 1/2 cycle of the curve defined by equation 28 . . . . . 1 0

7 Digital fan crossing sun in the precession case . . . . . . . . . 15

8 Definition of variables needed to solve the horizon sensor problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

9 Relationship between A tc and 6 . . . . . . . . . . . . . . . . . . . 22

10 Position of the momentum vector relative to the sun and the zenith vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

11 Position of a sensor Si relative to the aspect sensor in a spacecraft coordinate system . . . . . . . . . . . . . . . . . . . . 26

1 2 Relative positions of a sensor axis, the spacecraft z -ax i s , and the momentum vector . . . . . . . . . . . . . . . . . . . . . . . 26

13 Relationships of sensor axis Si with respect to the spacecraft z-axis and the momentum vector . . . . . . . . . . . . . . 27

14 Relationships of a sensor axis with respect to an external vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

iv

CONTENTS

Page

ENERGY AND MOMENTUM CONSIDERATIONS . . . . . . . . . . . . . . . . 1

MEASUREMENTS USING THE DIGITAL ASPECT SENSOR . . . . . . . . 7

Calculationof /B and 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Calculationof 4 and 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2

Calculationof &o and $,, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

MEASUREMENTS USING A HORIZON DETECTOR . . . . . . . . . . . . . 17

SPACECRAFT ATTITUDE DETERMINATION . . . . . . . . . . . . . . . . . 24

Error Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Sensor Position as a Function of Time . . . . . . . . . . . . . . . . . . . 26

APPENDIX A ....................................... 00

APPENDIX B ....................................... 00

APPENDIX C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00

APPENDIX D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00

APPENDIX E ....................................... 00

APPENDIX F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00

iii

Figure

List of Illustrations (Continued)

Page

15 Relationships necessary for finding times of maximum and minimum angles between a sensor and an external vector . . . 30

A Sun sightings defined within a quantized time base . . . . . . . . 00

AA Relationship of spacecraft to sunlit earth . . . . . . . . . . . . . .

l - ATTITUDE DETERMINATION OF SPIN STABILIZED

SPACECRAFT USING A DIGITAL ASPECT SENSOR

ENERGY AND MOMENTUM CONSIDERATIONS

The problem of the force free motion of a rigid body has been extensively treated by a number of people and a great many methods for solutions have been formulated. The most common and most useful for our discussion was formulated by Euler. The Euler angles are shown in Figure 1. They a r e defined as three successive rotations performed in specific sequence. The sequence is as follows: (1) Rotate the initial system of axes counter-clockwise about the Z axis by an angle 4 , which produces a set of intermediate axes (x', y ' , 2' ) . (2) Next rotate the intermediate set of axes (x' , y' , z ' ) counterclockwise through an angle 8, about the X' axis forming the line of nodes. This produces a second set of intermediate axes (x", y", z"). (3) Finally, rotate this set of axes counterclockwise through an angle 4 about the z" axis, forming the (x, y, z) axes.

The three angles t, , 4, and $ then completely describe the orientation of the x, y, z coordinate system with respect to the X, Y, Z coordinate system.

Now let the X, Y, 2 coordinate system define an inertial space. Assume a rotating satellite situated in this inertial space where the momentum vector, (L), of the rotating satellite is orientated along the Z axis of the inertial space. x, y, and z may now represent a coordinate system fixed in the rotating satellite. The axes x, y, z a r e aligned with the three principle moments of inertia of the satellites I,, I,, and I , . The instantaneous values of momentum about the x, y, and z axes become p,, p,, and p, respectively.

p x 1 = I ox

p = I G) Y 2 Y

1

P, = I3 wz

1

z \ i' Y

Figure 1 -Euler rotation angles

Where us, w y , and wz a r e the instantaneous values of angular velocity about x, Y, and z .

From examination of Figure 1 it can be seen that the components of L are:

(4) p, = L s i n 6 s i n $

( 5) p = L s i n B c o s $ Y

By the process of infinitesimal rotations' a relationship is established between the angular velocities ax, m y ,

Euler angles 6, 4, and 6 . These relations are stated in equations (7), ( 8 ) , and (9).

and w z , and the rate of change of the

w = qi s i n e s i n + t b cos +

w = d s i n e cos 4 - b s i n + Y

w, = c$ c o s 6 t $L

Substituting 7, 8, and 9 in 1, 2 and 3 gives

P, = I, (4 s i n e s i n + t i, cos +)

(11) p I, (6 s i n e c o s + - 6 s i n $) Y

' P g . 134, Goldstein, H. - Classical Mechanics, Addison-Wesley Publishing Co., Reading, Mass. 1957.

2

Multiplying (4) by sin # and (5) by cos # gives,

p, s i n IC, = L s i n 8 s i n 2 $

p c o s 1 L s i n e cos2 $ Y

Now add expressions (13) and (14) to get

Assume for simplicity that the satellite is balanced in such a way as to make - I, - I,.

Making this assumption, and substituting ( lo) , and (11) in (15) gives

(16) L s i n 8 = I , 6 s i n e

o r L A , ( $

Again, multiplying (4) by cos $ and (5) by sin $ and subtracting gives the following expressions.

p, c o s # - p Y s i n # = L s i n e s i n $ c o s $ - L s i n 8 s i n $ c o s $ (18)

p, c o s # - p Y s i n $I = 0 (19)

Substituting expressions ( lo) , and (11) into expression (19) gives

I , B = o

o r B is time independent.

3

Substituting expression (17), into expression (6) and then (6) into (12),

I, & c o s 8 = I, (4 c o s 8 + 4) (21)

I - I I3 $J

b, = (I, - 13) c o s H a) 4 = 1 3 &OS e

I3

4 is the angular velocity of the satellite about the satellite z-axis. This will be called spin rate. This is to be distinguished from the apparent rotation rate of the satellite with respect to a fixed external point. This apparent rotation rate has an average value of ( 4 + 4 ).

4 is the angular velocity of the satellite z-axis about the momentum vector (or more precisely, the angular velocity of the line of nodes). This will be called the precession rate.*

6 is the rate of change of the precession half-cone angle. Such a motion is called nutation.* A satellite with appreciable nutation presents an extremely difficult, if not insoluable, motion determination problem. Nutation is the result of I , + I , and e ' # o .

A s previously mentioned, if I , is made equal to I, , 6 = 0 and there is no nutation.

8 is the precession cone half angle.

A freely rotating satellite which has both spin and precession motions ex- periences time dependent accelerations at various points in its structure. Since most satellites a r e not absolutely rigid bodies, some flexing of the structure occurs and energy is thus dissipated through frictional losses. Momentum is,

*NOTE: The use of the terms precession and nutat ion are defined i n a force free f ie ld according to the astronomers' convention, i .e. : Precession rate i s the rate of rotat ion of the l ines of nodes and nutat ion i s the short term f luctuation in the precession cone ha l f angle. Th is convention i s qui te dif ferent from that used in the theory of gyroscopic motion. Gyroscopic precession i s the motion of the gyroscopic spin axis as a result of an external torque applied perpendicular to the spin axis. t ion i n the gyro sense i s similar to precession i n the astronomical sense.

Gyroscopic nutat ion i s the wobble of the gyro ax i s about the momentum vector. Nuta-

4

I'

of course, conserved as long as no external forces act on the satellite. The effect of such energy losses can be seen by examining the following equations:

The Lagrangian is written

Since we are considering force free motion, V (e, 4, $J) = 0 .

Substituting (7), (8), (9), in (23) gives

Assuming again that the satellite is balanced so that I , = I,, expression (24) reduces to

Since 8 is a constant (equation 19), e' = 0. Now substituting equation (22a) in (25) gives

Substituting L = I, 4 from (17) gives

5

In the absence of any external forces L is a constant. If precession motion causes energy to be dissipated internally through battery liquids sloshing, non- rigid members flexing, etc., T becomes smaller. Since L is not affected by internal dissipation of energy, the value of e then must change; as shown in Fig- ure 2, the value of 8 will tend toward O o , or 90°, o r remain constant depending on the value of the ratio I, /I,.

T t 0 eo- 90

@ CASE I 4 < I 8 TENDS TOWARDS 0'

@ CASE II i > l 8 TENDS TOWARDS 90' 13

T 1 0 80- 90

@ CASE 1 1 1 i = I T INDEPENDENT OF 8 13

Figure 2-Relationship of energy to precession half angle for constant momentum.

Case I - If I,/I, < 1, 8 tends toward Oo. In this case the precession-coning will damp out in time. Since it is usually desirable for the satellite to rotate about the z - axis, most spin stabilized satellites a r e balanced so that the z -axis coincides with the largest inertia moment, that is so that I , / I , < 1.

It should not be assumed that damping of the precession cone implies that & goes to zero. Equation 22 states that when e = 00, & = I,$/I, - I , . It can be seen from Figure 1 that the apparent rotation velocity of the satellite for 8 = 0' is (4 + $ ) *

Case I1 - On the other hand if I ,/I, > 1, B tends toward 90'. In this case the precession cone opens up until the satellite begins to spin like a propeller. In this mode $ goes to zero (Equation 22).

Case I11 - The third possibility is that I, /I, = 1. In this case the satellite is perfectly symmetrical about all axes and there is no way of distinguishing the z - axis dynamically. In this case I, = I, and from equation 22 again 4 = 0.

The final result of any of these three cases is to cause the satellite to rotate with no wobbling motion about the axis of the largest moment of inertia.

It should be noted that Case I11 is difficult to achieve in reality. I , is seldom made exactly equal to I, and in fact, our symmetry assumption that I, = I, is seldom precisely true either. Thus, it may happen that if the three moments

6

a re very nearly equal, the condition where I, > I, > I, , o r I, < I, < I, may inadvertently occur. This possibility raises a fundamental problem associated with asymmetrical rotating bodies.* That is, if a body is shaped so that either I, > I, > I, , o r I, < I, < I, , and the body is set rotating about I,, its motion is unstable. This instability is such that any slight perturbation will send the body into wild, unpredictable, and quite violent gyrations. The change in spin axis caused by this type of instability occurs in a time interval on the order of one revolution period and requires no energy dissipation. Thus, this instability is quite different from Case I1 where the change in spin axis occurs over many spin periods and depends entirely on the dissipation of energy.

MEASUREMENTS USING THE DIGITAL ASPECT SENSOR

To make the measurements necessary to determine the motion of a spinning satellite, assume that a photo detector is mounted behind a slit situated on the side of the rotating satellite. Let the axis of this detector be along the satellite x -axis with the slit positioned parallel to the satellite z-axis. Then the field of view of this detector will be a fan which will sweep across the sky, which is the celestial sphere, as the satellite rotates. This shown in Figure 3. Now further assume that this fan field of view is quantized into a number of segments as shown in Figure 4. This may be done if instead of a simple slit and photo

OF NODES /’ X

SATELLITE t AXIS

Figure 3-Celestial sphere representation of a precessing satellite using a digital aspect sensor.

Figure 4-Quantized f ield of view of a digital aspect sensor

NOTE: For a rigorous treatment of the stability of asymmetrical rotating rigid bodies see page 116 of “Mechanics” by Landau, L. D. and Litshitz, E.M. , Addison-Wesley Publishing Co., 1960.

7

detector, a digital aspect sensor2! view. Each of the fan segments has a unique identifying number. As the satellite rotates, if a bright object such as the sun should be in a portion of the sky swept out by one of these quantized fan segments, the number corresponding to that particular segment would be transmitted to the ground. This number then defines the angle between the satellite z -axis and the sun vector. This angle must lie between the upper and lower edges of the particular quantized fan segment. The angles between the edges of these quantized segments and the z -axis a re listed in a calibration table and are designated n l , a2, a s , . . . . . . . . . . . . u . If there is no precession coning, the same segment number will be repeatedly transmitted upon each revolution of the satellite.

is used to provide the fan shaped field of

Most of the discussion in the rest of this section deals with problems caused by precession coning. For the zero precession case the spin rate 4 and precession rate & can be found from equations (38) and (39). The reader may then skip to equations (51) and (52) where the initial phase angles $o and 4o can be set equal to zero at time t o . Much of the following discussion of precession motion determination will reduce to a degenerate case for zero precession.

Calculation of /j and 0

In the event of precession coning the motion of the fan across the celestial sphere is quite complicated. The first quantities to be calculated from the data a re the angles ,d and 0 .

If the satellite is balanced so that I, = I, the angle a between the z - a x i s of the satellite and the sun vector will be a function of time given by (28)

c o s a - c o s /3 c o s 8 + s i n B s i n ,b s i n 4 t (28)

Where P is the angle between the momentum vector and the sun vector. Figure 5 is a graph of (28) showing sun sightings for the two cases where I , > I, , and I, < I, . If I, > I, , the fan "sees" the sun many times for one cycle of 4. If I , < I, , completes more than one cycle per sun sighting. Often the best

2Albus, James S., *A Digital Solar Aspect Sensor,'' Astronautics, January 1962 also NASA TN D-1062.

3Albus, James S. and Schaefer, David H., "Satellite Attitude Determination: Digi tal Sensing and On-Board Processing," IEEE Transactions on Space Electronics and Telemetry, Vol. SET-9, pp. 71-77, September 1963.

8

. ,'

ONE PRECESSION

~ - 0- 180- 360- 540- +t

a', an a7 (16

a4 a3 a2 Q I

as

0- 360. 720- IOBOe 1440. #t

Figure 5-Typical plots of sun sightings from a dig i to I aspect sensor

method of calculating p and 8 is to plot the a limits for the sun sightings and f i t a curve. It is possible to apply a statistical approach to the calculation of ,B and e over a period which is long compared with any beat frequency set up between $ and $. However, this is sometimes in practice a prohibitively long period of time.

To calculate the values of ,B and 8, take some time interval T, - To such that at T, and To (4 - n / 2 ) < 5, and ( # t n / 2 ) < 5, where 5 , and 5 , are arbitrarily small angles. From Figure 3 it can be seen that these restrictions make a a minimum at To and T,. Between To and T, the sun appears at semi-regular

9

intervals which a r e not integrally related with 4. Therefore, over a period which is long compared to any beat frequency which might appear between & and 4, all values of (&t) between 0 and 2~ become equally probable.

Now the probability of a sun appearance in the segment nearest the spin axis is equal to Na /N (where Na is the number of appearances of the sun in the nearest segment, and N is the total number of times the sensor fan crosses the sun during TI - To). Thus, the probability at any instant of a > aa is Na I”.

A half cycle of the curve (28) is plotted in Figure 6.

a

I I I

If an arbitrary point along the & t axis is chosen, the probability that the corresponding value of a is less than aa is 2ta/Tp The relationship between aa and t a is given by substituting in (28)

c o s aa = c o s p c o s O t s i n p

s i n B c o s (6 ta )

Figure 6-Plot of 11’2 cycle of the curvedefined by Equation 28. which can be expressed as

We have already shown that the probability of a solar appearance during the interval TI - To in the nearest fan segment is N,/N. If aa is set equal to the value of the most distant edge of the nearest quantized segment, then Na/” may be substituted for 2 ta/T in (29b) to give (30).

Similarly,

10

I ’

t

where aa is the most distant edge of the nearest quantized fan segment to the spin axis, and Na is the number of sun appearances in the nearest segment. ab is the nearest edge of the most distant quantized fan segment. N, is the number of sun appearances in all segments except the most distant. N is the total number of all sun appearances during the interval TI - To.

The restrictions on To and TI may now be disregarded by making N sufficiently large.

Writing (30) and (31) in more convenient form for solution gives

ad - bc t a - c - a ( d t 1) - c ( b t 1 ) - d - b cos (P t 8 ) 1 -

a d - b c - a t c - a (d - 1) - c (b - 1) cos ( p - e) = - d - b d - b

where

a = cos aa

b = C O S ( n N a / N )

b c = cos a

d = cos ( n N b / N )

Note that equation (32) is ambiguous a s to the quadrant of ( p + 8) and (p - 8). Thus the digital sun sensor does not indicate directly whether the sun is inside or outside of the precession cone. Ignorance of this ambiguity can lead to con- siderable confusion if the sun happens to lie inside the cone. Resolution of the ambiguity is not straightforward unless the approximate size of the precession cone can be independently measured.

If the rotation rate can be determined by an independent means such a s a magnetometer or a telemetry AGC record use of equations (36) and (37) will provide an indication of whether the sun is inside o r outside the precession cone. However, if the magnetic field vector or the ground receiving station is inside the precession cone this method itself is subject to the same type of ambiguity. Another method is to reconstruct a time history of a precession cycle on the celestial sphere. The phase of the 4 and IC, angles will usually provide an indication of whether (,B - 0) is plus or,minus.

11

Where the sun only appears in two quantitized segments, there are not enough known quantities to solve for both ,8 and 8. In this case, unless 8 can be estimated from previous data, assume 8 equal to width of smallest segment in which the sun appears. Then solve (33) for p .

a b k c J(b2 + c 2 ) - a2

(b2 t c 2 ) cos p = (33)

where

a = c o s aa

b = C O S 8

c = s i n 0 cos ( n N a / N )

In this case, if B is constant over a period of days, an accurate measure of H can sometimes be made by noticing how long the sun's appearances oscillate from one segment to the other. The velocity of the sun's yearly motion, multi- plied by the crossover oscillation time, give 2 8. ,8 at the midpoint of the crossover time is exactly the value of the calibrated angle an. Other factors besides the sun's apparent motion can of course cause p to slowly vary over a period of days. These factors a re external torques caused by magnetic fields, radiation pressure, aerodynamic drag, plasma winds, gravitational gradients, etc.

Calculation of and

¶

Having now calculated 0 and p we proceed to determine the spin rate II, , and the precession rate & . In the general case where precession motion is present, the apparent rotation period, which is the time between successive crossings of the sun by the fan field of view, will not be uniform. This is due to the fact that sometimes the fan is tilted forward when it crosses the sun, and sometimes backward. However, the average value of the apparent rotation period is constant. In fact, 4 and 4 may be determined from the average apparent rotation period by the following method:

#

12

Choose X in Figure 3 so that the sun lies in the XOZ plane. This may be done with no loss in generality. Choose a time that the fan crosses the sun such that (6 - 7r/2 ) < 6, and ( + + 7r/2 ) < c2 where El and C2 a re arbitrarily small angles. Call this time To. This condition can always be satisfied at some time since 4 and 4 a re not integral multiples of each other. Now wait for another time when the fan crosses the sun such that these same restrictions can be applied to 4 and $. Call this time T, . Count the number of times the fan crossed the sun between To and T, . Call this N r .

Then the average period

Now from To til number of degrees).

and during that same

Tl - T o T r- N a v

T, , $ has made N revolutions (within an arbitrarily small Therefore,

+

27rN+ = 4 (T, - To); (34)

time 4 made N4 revolutions. Therefore,

27rN+ = 4 (T, - To). (35)

Now if I, > I, the value of N+ can be immediately determined by inspection of a plot similar to Figure 5. In fact in the case where I, > I , , if one has a rough knowledge of I,/I, , N+ can usually be found by inspection. However, if there is any doubt about N+, or if the sun lies inside of the precession cone a more rigorous approach is more reliable.

Assume for our discussion a fan field of view 180' in length and negligibly thin. In Figure 4 this would mean that a, = 0 and a, = 180'. From Figure 3 we can write

and

The case where Nr > N+ in equation (37) arises when the sun lies inside certain regions within the precession cone. These regions are quite complicated and will not be discussed here.

For the case where the fan is not 180' in length (i.e. a, > 0 and/or un < 180') equation (36) holds where (0 t a l ) < p < (an- 8) and equation (37) holds where (0 - a,) > ,h > value for Nr can sometimes be determined by inspection of a plot similar to Figure 5.

[360° - ( an + S)] . If these conditions are not satisfied, a

If the conditions for equation (36) hold, it can be combined with equations (22), (34), and (35) to give equations (38) and (39).

2 T N ~

and

2 n N

(38)

(39)

Now i f the restrictions on choosing the times To and T, are discarded, the uncertainty in (6 + 6) will be less than f (50%/Nr ). Therefore, i f To and T, are merely chosen a large number of revolutions apart so that N > 50 (38) and (39) give quite accurate answers.

The accuracy of 4 and 6 from equations (38) and (39) depends, of course, on the accuracy of 11/13 and 0 . If 4 and 6 can be roughly determined from equations (38) and (39), and then the knowledge of 4 and $ used to find N+ by inspection, equations (38) and (39) can then be solved again with 4 and $ as known and 11/13 unknown. This kind of iteration can be used to refine the value of I , / I 3 and even under certain conditions to determine 8 .

Calculation of +o and $o

The integrals of motion 4, & , B having now been calculated, it is necessary to establish initial conditions in order to define the instantaneous position of the

14

satellite in inertial space at a particular time. A particular passage of the slit across the sun may now be chosen as occurring at to.

In Figure 7 the digital fan is shown crossing the sun. aa and ab correspond to the angles between the spin axis and the upper and lower edges of the quantized fan segment containing the sun. From this figure the following equations may be written

c o s a = cos ,B cos 8 + sin /3 sin 8 cos (90" - 4)

o r in the case of ua and ab

cos aa = cos /3 c o s 6 + sin p sin 6 c o s (90" -

cos ab = cos /3 cos e + sin /3 sin 8 c o s (90' - +B)

Substitute the relation

4 = 6 ( t - t o ) t 5b0.

c o s aa = c o s p cos 8 t sin /3 sin 0 sin (4 ( t - to) t q50A)

C O S ab = C O S /3 c o s e + sin ,8 sin 8 sin (4 ( t - to) t boB)

(43)

(44)

(45)

Figure 7-Digital fan crossing sun in the precession case

15

' I

At t = to (44) and (45) can be solved for +OA and +OB. +o must lie some- where between +OA and 40B. However, the solution of (44) and (45) is ambiguous regarding the quadrant of +OA and +OB. This ambiguity may be resolved by choosing a value of t at a sun sighting approximately an odd number of quarter periods of C$ away from to . Solve (44) and (45) again for cos ua , and cos ab. This time try first one possible quadrant value for +OA and +OB and then the other. Compare the two values of cos ua and cos ab obtained with the cosine of the actual value of ua and u

indicate the proper quadrant for +OA and +OB.

measured at time t . The closest value should

Having now determined unique values for +OA and a,, ,.it is possible to narrow the limits on the value of +o . The precession rate 4 is accurately known, and the time of each sun sighting is also well known. Therefore, equations (44) and (45) can be repeatedly solved for +OA and +OB by inserting ua , a b , and t for each sun sighting after t o . Since +o must always lie between +OA and 4" must lie between the maximum of the smaller values and the minimum of the larger values of the pair +OA and +OB . This is somewhat tricky when put into practice since whether is the larger of the pair depends on the quadrant of the argument [& (t - t o ) + +01 . If cos [& ( t - t o ) + +01 is positive, +oA > +OB , but if COS [&( t - t o ) + 401 is negative, +OA <

+OA or +OB

Once +o is determined accurately enough, solve (47) for cos u0.

c o s a. = c o s p c o s e + s i n p s i n e c o s (90" - +o)

where 0 5 a0 < 180'

Now from the same spherical triangle in Figure 3

s i n p cos +o

s i n a cost+ho

0

c o s p - c o s 0 cos uo sint+ho =

s i n e s i n a.

Solve (49) and (50) for $o.

16

(47)

(49)

(50)

The instantaneous value of $J and can now be determined by

These equations will hold for some length of time t - to . When e r ro r s begin to accumulate, a new t o should be chosen and a new set of initial conditions computed.

If the time base in the spacecraft is quantized, there is a quantizing under- tainty in the value of the value of t o under the conditions of B = 0 are contained in Appendix E.

to . Equations for removing quantizing uncertainties in

MEASUREMENTS USING A HORIZON DETECTOR

Thus far all the measurements have been made on the sun. Since the sun is considered a point source at infinity, there is an axis of symmetry from the center of gravity of the satellite to the sun. Another inertial reference point is necessary in order to eliminate the ambiguity caused by this symmetry. In general what is needed is another known vector which is not colinear with the sun vector, and an angle 6 which is measured from this second vector to the spin axis. This second vector may correspond to the magnetic field, to the subsatellite zenith vector, o r any other desirable vector.

In the following discussion the use of horizon sensors to measure an angle 6 between the spin axis and the subsatellite zenith is considered. However, the equations in Section IV apply as well to a 6 angle measured by any other type of sensor from any other type of second vector.

Infra-red horizon sensors which detect the temperature discontinuity caused by the warm earth against the cold background of outer space can provide horizon sensor data on the dark side of the earth as well as on the sunlit side. Visible light horizon detectors can be used when data is needed only on the sunlit side. Conditions when visible light detectors can be used a re in Appendix F. Either type detector usually has a pencil beam field of view arranged so as to provide a pulse upon crossing either into or out of the earth's disc. Sometimes it is

17

necessary to transmit the knowledge of whether a pulse represents an "in" or crossing, but for satellite applications the difference can usually be deter-

mined by relative pulse spacings o r prior knowledge about the approximate spin axis orientation.

lb l

Figure 8-Definition of variables needed to solve the horizon sensor problem

Information is contained in the repetition rate of the pulse pairs, in their spacing, and in their position relative to the sun appearances in the quantized fan. The repetition rate of the pulse pairs can provide a measure of the rotation rate. The information contained in the pulse spacing provides a measure of 6 , which in this discussion is the angle between the spin axis and the subsatellite zenith vector. The subsatellite zenith vector is a vector from the center of the earth through the center of gravity of the satellite.

If the earth is considered to be a disc inside the celestial sphere, where the center of gravity of the satellite is the center of the sphere, an expression can be derived relating the pulse spacings to b .

Figure 8 represents a satellite rotating with no precession so that B = 0. An earth horizon pipper is mounted at an angle from the satellite 2 - axis so as to sweep out a cone which cuts the earth horizons at A and C. From Figure 8b it can be seen that

- = c o s EA

where A t , is the time between horizon pulses and T is the rotation period. From the figure, various other relationships can be derived:

- - ED = E t a n (90' - a ) where OE = O H c o s y

18

(53)

(55)

- - DB = = c o s (90' - 6) where OG = OH c o s p

- E A = m s i n y

Substituting all these values in expression (53), gives

which reduces to

cos p .+ c o s y cos 8 = s i n y s i n 8 cos - (YA) where

rE s i n p =- rE t h

where

rE = mean radius of earth

h = apparent altitude of satellite

Rewriting so as to solve explicitly for 8

- d e f f /(e2 + f2) - d2 c o s 6 =

(e2 t f2)

(57)

(59)

\

19

where

d = c o s p

e = c o s y

f = s i n y c o s (nAt,/T)

Expression (61) is plotted in Appendix A. It double valued for most of the important cases. In order to minimize this ambiguity it is usually best to make y = p or 180° - p . In order to remove the ambiguity altogether a second horizon sensor mounted with a different y angle can be used. If two sensors are used it is usually best to mount one at an angle y, = p and the second at an angle y, = 180' - P . With a system using two such sensors, the absence of a signal from one o r the other of the sensors usually will eliminate the ambiguous solution from (61). When both sensors scan the horizon the solution for 8 becomes single valued, and in addition the satellite apparent altitude can be calculated.

nn t , c o s p + c o s ' y , c o s b T s i n 'y, s i n 8 c o s - (624 T

n A t , (62b) c o s p + c o s y , c o s 8 = s i n y s i n 8 C O S -

b T

where A t , is the pulse spacing from sensor A and A t , from sensor B. Subtracting (62b) from (62a) and reducing gives

is the pulse spacing

P - 9 t a n 8 E-

l - m

where

p = cos y ,

q - c o s ' y , -

1 = s i n 'ya c o s (nAt , /T )

m - s i n ;yb c o s ( n A t b / T ) -

20

Solution of equation (63) is plotted in Appendix B.

Substituting the values of 8 just calculated back into equation (62a) and solving for p gives

7T 0 < P < - 2

Rewriting equation (60) gives the apparent satellite altitude

This apparent altitude may differ from the true altitude by the thickness of the atmosphere.

Information determining the value of 8 is also contained in the relative position of the horizon pulses to the quantized fan crossing of the sun. If the satellite is spinning so that 0 = O O , ' horizon pulses will be symmetrical about the time at which the quantized fan plane crosses the center of the earth. Fig- ure 10 illustrates the case where the horizon sensors a re mounted so that their fields of view lie in the plane of the quantized fan. The center of the earth is located on the celestial sphere at (RAZ + 180') and (-Dz), where RAZ is the right ascension and DZ the declination of the subsatellite-point zenith vector. (180'-7) is the great circle a rc from the sun to the earth's center on the celestial sphere, where 7 is the great circle arc from the sun to the subsatellite zenith vector. A t c is the time midway between horizon pulses minus the time the quantized fan crosses the sun. The relationship between 6 and A t c can now be written from the spherical triangles in Figure 9.

c o s (180O - 7 ) = c o s ,b cos (180O - 8) + s i n ,B s i n (180" - 6 ) (66)

2 1

' w

Figure 9-Relationship between Atc and b

Figure 10-Position of the momentum vector relative to the sun and the zenith vector

or

c o s 7 = c o s ,B cos 6 - sin ,B sin 8 cos [(& t 6) btc]

Solving for 8 gives

where r = C O S 7

s = c o s p

u = -sin ,B c o s (27rAtc/T)

0" f S < 180"

22

I (

and where

cos = sin DZ s i n Ds + cos DZ cos Ds cos (RAs - RAZ)

where

D = declination of subsatellite zenith Z

Ds = declination of sun

RA = right ascension of subsatellite zenith

RA = right ascension of sun S

Calculation of 8 by (68) should result in 2 solutions. This twofold ambiguity may be resolved by choosing the solution of (68) which most closely corresponds to values of b obtained from (61) or (63).

Calculation of 6 is often in practice a difficult task owing to the nature of the horizon. Pulse pairs do not always indicate the true horizon. In addition the apparent altitude of the satellite must be measured not from the earth's sur- face, but from some vaguely defined part of the atmosphere which is often dependent on weather conditions. Finally the earth is not a perfect sphere.

Calculation of 8 using equation (63) eliminates consideration of the apparent altitude of the satellite, and has proven in practice to usually provide the most reliable method of calculating. Use of equation (68) also eliminates consideration of apparent altitude.

Perturbation analysis of the solution for 8 indicates that under certain conditions one type of solution for 6 is less sensitive to measurement e r ro r s than another. This type of perturbation analysis is perhaps most understandable when done graphically. Appendix A, By and C contain plots of the solutions of equations (61), (63), and (68) under various conditions. Sensitivity of 6 to perturbations in various other parameters can be examined on these plots.

23

If there is some precession motion present so that B # 0, some sor t of averaging procedure is needed to obtain an average 6 . This average 8 then will be the angle between the second known vector and the satellite momentum vector.

Sufficient information has now been obtained to calculate the instantaneous orientation of the satellite for any moment of time. Thus, knowing the mounting arrangements of any sensor on the satellite, it is possible to calculate the sensor orientation for any time.

SPACECRAFT ATTITUDE DETERMINATION

The spacecraft momentum vector position can now be calculated. From Figure 10 we may write equations for the values of the auxiliary angles L1 and L3.

c o s /? - c o s q c o s b c o s L1 - s i n 71 s i n 6

s i n ,h s i n L2 s i n L1 x s i n 77

where

s i n Ds - c o s q s i n DZ s i n q c o s DZ

cos L3=

s i n (RAs - RAZ) C O S Ds s i n 7j

s i n L3 x (73)

Now using the values of L1 and 13 from (70-73) we may find RA, and DL the right ascension and declination of the momentum vector.

24

L'

s i n DL = s i n DZ c o s 6 t cos DZ s i n 8 c o s (L3 -t 1 1 ) (74)

n < n -- < DL - - 2 2

C O S 6 - s i n DZ s i n DL cos DL c o s DL

C O S (R% - RAZ) =

s i n (L3 + 11) s i n c o s DL

s i n (RA, - R A Z ) =

(75)

RA, = RAz + (R% - RAZ) (77)

This solution is unique. If the quadrant of (6 + 4) A t c is unknown, L1 must be found from equation (70) and two solutions result.

Er ror Calculations

It is well to question at this point the accuracy of the position of the momentum vector determined by these methods. Tables of values of the e r ro r in the position of the momentum vector caused by a one degree e r ror in 6 is contained in Appendix D. From these tables certain facts become obvious.

The e r ror is large whenever 77 2 0, or 7 2 180'. The e r ro r is also large whenever 77 2 + 6 , 6 2 77 + p , ,b 2 6 + 7 7 , or 77 + ,8 + 6 2 360'.

Also contained in Appendix A, B, and C are a ser ies of plots of 6 vs. A t a , A t , , and A t c under various conditions. From these plots then it is possible to estimate the e r ro r in momentum vector position caused by an er ror in the measurement of A t a , At,, or A t c . From the plots find the slope of the curve under the conditions in question, multiply it by the uncertainty in A t a , This gives the uncertainty in 6 . Then go to table and look up the e r ro r in momentum vector position corresponding to the e r ro r in 6 .

At,, A t c .

25

The table in Appendix D is symmetrical and therefore the e r ro r in momentum vector position caused by an uncertainty in p can also be obtained by interchang- ing the values of p and b and assuming that 6 is the uncertain quantity. Thus the uncertainties from f i can then be combined with those from b to give the tot a1 uncertainty .

SPACECRAFT Z-AXIS

J / ASPECT SENSOR

AXIS

A Z S E N i

Si th SENSOR

AXIS

-

'\ ' \ I

\ I \

J

Figure 11-Position of a sensor Si re lat ive to the aspect sensor i n a spacecraft coordinate system

Sensor Position as a Function of Time

The momentum vector having been determined, the constants of motion calculated, and the initial conditions at time to measured, it is now possible to reconstruct the motion history of the spacecraft both forward and backward in time from to over an interval a s long a s theuncertainties in the above quantities will permit. The look direction of any sensing device on the spacecraft can then be found at any time if the sensor position relative to the aspect sensor is known.

Assume a sensor Si mounted on the spacecraft so that its axis makes an angle ARCi with the I, inertia axis, which is the satellite z-axis, and so that the sensor axis lies at angle AZSEN, from the aspect sensor axis measured east along the satellite equa- tor. See Figure 11.

The following solution for the instantaneous position of the individual sensors does not use the conventional coordinate transformations by matrix multiplication. The matrix method is perhaps more elegant, but in general the number of com- puter operations for the matrix method is considerably in excess of that needed for the method described below.

NORTH

T NSOR

AXIS

Figure 12-Relat ive posit ions of a sensor axis, the spacecraft z-axis, and the momentum vector

26

Figure 13-Relationships of sensor axis Si with respect tothe spacecraft z-axis and the momentum vector

Define an angle A as shown in Figure 12.

7T A - Q + - - + 2

where 4 = 4 ( t - to ) + +o as defined in equation (52) and where

Also from Figure 12

s i n (RAs -RA,) C O S Ds

s i n P s i n Q =

s i n Ds - c o s p s i n DL s i n ,B c o s DL

cos Q =

(79)

where $J = $ ( t - to ) + $,, as defined in equation (51)

27

Now solve for the auxiliary angles Ei and C, from Figure 13. If 8 = 0

E, =ARC, and Ci = n - B i

However, if U > 0

cos E. z c o s h' cos ARC, -t sin H sin ARC, cos Bi

sin Bi - s in ARCi sin Ei

sin Ci

cos ARC, - cos B C O S Ei

sin H sin E, COSC, :

Finally the right ascension RA, and declination Di of the sensor Si can be found by

sin Di = cos Ei sin D, + sin Ei cos D, cos (A - Ci) (85)

- - - < D i n < n - - 2 2

where D, is the declination of the momentum vector from equation (74), and

c o s E - sin D, sin Di C O S (RA, -RA,) = i

cos D, cos D,

sin (A - C,) sin Ei

cos Di sin (RA, - RA,) = (87)

28

RA, (RA, - R%) + RA, (88)

where RA, is the right ascension of the momentum vector from equation (77)

Thus the aspect problem is uniquely solved for any instant of time. It is now possible to obtain the angle between the sensor Si and an external vector such as the magnetic field vector or velocity vector. The coordinates of the Si vector need only be substituted into equation (89) along with the coordinates of the external vector to obtain

cos Ri = s i n Di s i n Dx + cos D, cos Dx cos (RA, - RAx) (89)

0 5 . 0 , < 7 r

where 0, = the angle between the external vector and Si at an instant of time.

Dx = declination of the external vector

RA = right ascension of the external vector X

Of course a great deal can be known about the behavior of a particular sensor with respect to an external vector without the complete solution. For instance if the momentum vector is known, the minimum angle a sensor ever makes with an external vector can be found. From Figure 14 the minimum angle Si makes with an external vector is given by

R, min = 10, - ARC,

R, inin = 0

- e i f ( I R , - A R c ,

i f (InL - ARC, I - B

- 6 2 0 ) (90)

where .QL is the angle between the momentum vector and the external vector, and 9 is the half angle of any precession which may be present.

29

Figure 14-Relationships of a sensor axis with respect to an external vector

NORTH

Figure 15-Relationships necessary for finding times of maximum and minimum angles between a sensor and an external vector

In like manner, the maximum angle a sensor makes with an external vector is given by

J

R, max = 0, t ARC, t 8 i f (0, t ARC, t 69 5 TT

i f (0, t ARC, t 8) > n

a n d (12, + ARC, - 8) > TT

R max = 271 - (0, t ARC,) t e i

i f (0, t ARC, t 8) > n

and (0, t ARC, - 8) < n

n max = TT i

30

The time at which Ri m i n and R. I m a x occur can also be found rather easily in the case where B = 0. From Figure 15 find

sin Dx - c o s R, sin D, sin R, cos D,

c o s x =

cos Dx sin (RAx - RA,) sin R,

sin X =

(94)

(95)

The time at which a sensor makes a minimum angle with the external vector is then

where t, is the time at which the aspect sensor crosses the sun, and Q is the solution to equations (79) and (80).

The time at which a sensor makes a maximum angle with the external vector is

If B # 0, the general method of solving for Ri must be used a number of times in a roll period and the minimum value of 6, Similarly for Ri max . set equal to 0. . .

1 m i n

31

i

APPENDIX A

Plots of A t , and h Solutions of Equation (61)

33

DELTA V.S. SUN EARTH PULSE SPACING

9 x 360°

34


w.. . .-

20.. -

10.. -- I l l I - 1 i

'. 120. I

I---

. ~

1

- 160.

-

< < < < < < < - .

4- I

I

atr x 360" T

35


D ' E

L T A

9 x $60°

D

36

I '


9 x 360'

37


20. 40' 60. 80. 100. 120. 140. 160. 180. 200' 220' 2

x 360" T

38


x 360" T

39


x 360" T

40


x 360' T

41


x 360" T

42


&I x 360" T

43

DELTA V.S. SUN E A R T H PULSE SPACING

x 360' T

44

.*


9 x 360"

45


x 360' T

46


&A x 360' T

47


I xi IH I

x 360" T

48


&A x 360" T

49


9 x 360"

50

APPENDIX B

Plots of A t , and A t , Solutions of Equation ( 6 3 )

51

DELTA FROM A P A I R OF EARTH SENSORS

52

D E L T A FROM A P A I R OF E A R T H SENSORS

53

*


54

I ~

I

D E L T A


' r


180'

X 360'

56


57

3


30.

58

I f


59

I c

APPENDIX C

Plots of A t , and P Solution of Equation (68)

6 1


- ~360' T

62


- A'cx3600 T

63


x 360" T

64

I E


etc x 360" T

65

DELTA V.S. SUN EARTH PULSE SPACING 180'

170'

160'

I so<

140'

130

120

I10

100

90

80

70

6C

x

4c

3c

2C

IC

I !

20. 40.

z

&?L x 360° T

66

I P

DELTA V.S. SUN EARTH PULSE SPACING 180

I70

160

I50

140

I30

120

I10

D 100

E

L 90

T

A 80

70

60

40

30 i

20

10

C

A t c - x 360" T

67

180.

170'

160'

150'

140'

130'

120'

I IO'

D 100'

E

L 90'

T A 80'

70'

60

w

40

30

20

10

C 20. 40. 60.

.1


x 360° T

68

1. 260. :

DELTA V.S. SUN EARTH PULSE SPACING -

x 360" T

69

70

,

7 1

t

1 1 1 ' 1 I I I I I

72

73

74

75

I c

DELTA V.S. SUN EARTH PULSE SPACING Inn..

9 x 360"

76

77


- x360° T

78

79

4

80

8 1

4

.I I

I I I I I

82

83

/-

I

85

c 4

J

87

- 1 I I

88

EA TH I PUI SE

89

I

90

91

. L

4 .s. U N ~ E A I ~ T H " " 1 1

I ! I

I

PUqSE ISPAPIN

I i - l l

92

I

93

EAfiTH /-PUtSE

! I

' I - 1 - - - - . 1 - -

! I ' I ; . .

! I

........ I I , .

I -

I

94

APPENDIX D

Tables of Error in Momentum Vector Position Caused by 1" Error i n

6 for Various Values of and p .

95

O E L l A

2.0 4.0 6.0 8.0

10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 2 6 . 0 28.0 30.0 32.0 34.0 36.0 38.0 40.0 42.0 44.0 46.0 4 8 . 0 50.0 52.0 54.0 56.0 50.0 60.0 62.0 64.0 66.0 60 .0 70.0 72.0 74.0 76.0 78.0 80.0 8 2 . 0 01.0 86 .0 88.0 90.0 92.0 94.0 96.0 98 .0 100.0 102.0 104.0 106.0 108.0 110.0 112.0 114.0 116.0 118.0 120.0 122.0 124.0 126.0 128.0 130.0 132.0 134.0 136.0 138.0 140.0 142.0 144.0 146.0 148.0 150.0 152.0 154.0 156.0 158.0 160.0 162.0 164.0 166.0 168.0 170.0 172.0 174.0 176.0 178.0

E T L - 10.0 B E l A

IO 20 30 40

1.0 1.0 1.1 1.1 1.2 1.3 1.4 1.0 2.6 6 . 2 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0. 0 . 0. 0. 0. 0 . 0 . 0. 0. 0. 0.

0. 0. 0 . 0. 0. 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0. 0. 0 . 0 . 0. 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0. 0. 0 . 0 . 0 . 0 . 0 .

n.

0 . 0 . 0 . 0 . 0 . 2.4 2 .1 2.0 2.0 2.0 2.2 2.5 3.0 4.5

39.6 0. 0. 0. 0. 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0 . 0 . 0 . 0 . 0 . 0. 0. 0. 0. 0. 0 . 0 . 0 . 0 . 0 . 0. 0. 0 . 0 . 0. 0. 0. 0 . 0. 0. 0 . 0. 0 . 0. 0 . 0 . 0. 0 . 0 . 0 .

0 . 0 . 0. 0. 0 . 0 . 0 . 0 . 0 . 0 .

3.2 2.9 2.9 2.9 3- 1 3.5 4.2 6.2

59.5 0. 0. 0. 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0. 0 . 0 . 0 . 0 . 0. 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 .

3.0

0. 0 . 0 . 0 . 0 . 0 . 0. 0. 0. 0 . 0 . 0. 0 . 0 .

10.1 5. I 4.2 3.0 3.7 3.7 3.9 4. 3 5.2 7.1

79.4 0. 0. 0. 0 . 0. 0. 0. 0 . 0 . 0. 0 . 0. 0 . 0. 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 .

50

0. 0 . 0 . 0 . 0 . 0 . 0 . 0. 0. 0 . 0 . 0 . 0 . 0 . 0. 0. 0 . 0 . 0 . 0. 6 . 3 5. I 4 . 6 4.4 4.4 4 .6 5. I 6.1 8 .9

99.2 0. 0. 0. 0 . 0 . I 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 .

60

0. 0 . 0. 0 . 0 . 0 . 0 . 0. 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0. 0. 0. 7.2 5.8 5.2 5.0 5.0 5.2 5.7 6.8 9.9

119.0 0. 0. 0. 0 . 0 . 0. 0 . 0. 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0. 0 . 0. 0. 0 . 0 . 0. 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0. 0. 0. 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 .

7 0

0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0. 8.0 6.4 5.7 5.5 5.4 5.6 6. I 7.3

10.6 138.5

0. 0. 0. 0. 0 . 0 . 0 . 0 . 0 . 0. 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0 .

00

0. 0 . 0. 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0. 0 . 0. 0 . 0 . 0 . 0. 0 . 0. 0 . 0. 0. 0 . 0 . 0 . 0 . 0 . 0 . 0. 0. 0. 0.5 6.7 6 . 0 5.1 5.7 5.9 6.4 1.5

11.0 157.2

0. 0. 0 . 0 . 0 . 0 . 0 . 0. 0 . 0. 0. 0. 0. 0 . 0. 0 . 0. 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0. 0. 0. 0. 0. 0. 0. 0 . 0. 0 . 0. 0 . 0. 0 . 0. 0 .

9 0

0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0. 0. 8.7 6.9 6.1 5.0 5.0 5.9 6.4 7.6

11.0 169.2

0. 0 . 0 . 0 . 0. 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0. 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 .

LOO

0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0. 0. 0. 0 . 0. 0. 0. 0 . 0 . 0 . 0 . 0 . 0 . 0. 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0. 0. 0. 0. 0.7 6.8 6 - I 5.0 5 .1 5.0 6.3 7.4

10.7 157.5

0. 0. 0. 0 . 0. 0 . 0. 0 . 0. 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0. 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 .

110

0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0. 0 . 0 . 0 . 0. 0 . 0. 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0. 8.4 6.6 5.8 5.5 5.4 5.5 6.0 7.0

10.1 138.0

0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 .

120

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119.3 0. 0. 0. 0 . 0. 0. 0 . 0 . 0. 0. 0 . 0 . 0. 0. 0 . 0 . 0. 0. 0. 0 . 0. 0. 0 . 0 .

130

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0 . 0 . 0 . 0 . 0. 0. 0 . 1 .o 5.5 4 . 0 4.5 4 .4 4.5 4.0 5.5 1 .9

99.6 0. 0. 0. 0 . 0. 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 .

n.

140

0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0. 6.0 4.7 4.1 3.8 3 .1 3.7 4 . 0 4.5 6.4

79.7 0. 0. 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 .

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59.9 0. 0. 0. 0 . 0 . 0 . 0 . 0 . 0 .

160

0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0. 0. 3.5 2.7 2.3 2.1 2.0 2.0 2.0 2.2 2.9

4 0 . 0 0. 0. 0 . 0 .

170

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96

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62.0 64.0 66.0 6H.O 70.0 72.0 74.0 76.0 78.0 80.0 82.0 8 4 . 0 86.0 88.0 90.0 92.0 94.0 96.0

100.0 102.0 104.0 106.0 108.0 110.0 112.0 114.0 116.0 118.0 120.0 122.0 124.0 126.0 128.0 110.0 132.0 134.0 136.0

140.0 142.0 144.0 146.0 148.0 150.0 152.0 154.0 156.0 158.0 160.0 162.0 164.0 166.0 168.0 170.0 112.0 114.0 116.0

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97

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100.0 102.0 104.0 106.0 108.0 110.0 112.0 114.0 116.0 118.0 120.0 122.0 12+.0 126.0 128.0 130.0 132.0 134.0 136.0 138.0 140.0 142.0 144.0 146.0 148.0 150.0 IS2 .0 l S 4 . 0 lS6.0 I S 8 . 0 160.0 162.0 164.0 166 .0 168.0 110.0 172 .0 114.0 176.0 178.0

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30 40

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SO 60

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70 80 9 0 100 110 120 130

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lS9.8 2.3 1.8 1.6 0. 2 . ) 1.8 1.6 0. 2.8 1.8 1.) 0. 3.4 1 .9 1.5 0. s.0 1.9 1.5 0. 139.9 2 .0 1.s 0. 0. 2.2 1 .5 0. 0. 2.4 1.6 0. 0. 2 .9 1.6 0. 0. 4.3 1.6 0. 0 . 120.0 1.1 0. 0 . 0 . 1.8 0 . 0 . 0 . 2 .0 0. 0 . 0 . 2 .6 0. 0. 0. 3.4 0. 0. 0. 100.0 0. 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0. 0 . 0. 0 . 0. 0 . 0. 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0. 0 . 0 .

140 IS0 I60

0. 0. 0 . 0 . 0. 0 . 0. 0. 0 . 0. 0. 0 . 0. 0. 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0. 0. 0. 0. 0. 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0. 0. 0. 0 . 0 . 0 . 0. 0. 0 . 0. 0 . 0 . 0 . 0. 0. 0 . 0 . 0. 0 . 0. 0 . 0. 0. 0 . 0 . 0. 0. 0 . 0. 0 . 0 . 0. 0 . 0 . 0 . 0 . 0. 0. 0 . 0. 0. 0. 0. 0. 0 . 0. 0 . 0 . 0. 0. 0 . 0 . 0. 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0. 0. 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0. 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0. 0 . 0 . 0 . 0 . 0. 0. 0 . 0 . 0. 0 . 0. 0 . 0 . 0 . 0 . 0. 0. 0. 0 . 3.8 0. 0. 2 . 9 0. 0. 2.4 0. 0. 2 .2 0. 0. 2 .0 0. 0. 1.8 3.2 0. 1.7 2.4 0 . 1.6 2.1 0. 1.6 1.8 0. 1.s 1.1 0. 1.5 1.5 2.5 1.4 1.s 1.9 1.5 1.4 1.6 1.4 1.3 1.4 1.4 1.3 1.3 1.3 1.2 1.2 1 . 3 1.2 1.2 1 . 3 1 .2 1.1 1.3 1.1 1.1 1.3 1.1 1.0 1.3 1.1 1.0 1.3 1.1 1.0 1.3 1.1 1.0 1.3 1.1 1.0 1.3 1.0 1.0 1.4 1.0 1.0 1.s 1.0 1 . 1 1.7 1.0 1.2 2.3 1.0 1.7

(10.0 1.0 40.0 0 . 1.0 0 . 0 . 1.0 0 . 0 . 1.0 0. 0 . 1.0 0 .

110

0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0. 0 . 0 . 0. 0. 0 . 0 . 0 . 0. 0. 0 . 0 . 0 . 0 . 0. 0. 0. 0 . 0. 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0. 0. 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0. 0. 0 . 0. 0 . 0 . 0 . 0. 0. 0. 0. 0 . 0 . 0. 0. 0. 0. 1 .1 1.3 1.1 1.0 1.0 1.0 1 . 1 1 .2 1.6

20.0 0. 0. 0. 0 . 0 . 0. 0 . 0 . 0 .

98

,

E T A = 40.0 B E T A

10 2 0 30

0 . 0 . 0. 0. 4.0 1.5 1.2 1.1 1.1 1 .o 1.0 1.0 1.0 1.0 I .o 1.0 I .o 1.0 1.0 1.1 1.1 1.1 1.1 1.2 1.2 1.3 1.3 1.4 1.5 1.6 1.8 2.0 2.5 3.8

59.9 0 . 0 . 0 . 0. 0. 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 .

40 50 60 70 80

0 . 0 . 0. 0. 0 . 0 . 0. 0. 0 . 0 . 0 . 0. 0. 0 . 0. 0. 0. 0. 0. 0. 3.5 2.7 2.4 2.1 2.0 1.9 1.8 1.1 1.1 1.6 1.6 1.6 1.6 1.6 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.6 1.6 1.6 1.6 1.6 1.6 1.1 1.1 1.8 1.8 1.9 2.0 2.1 2.3 2.5 2.8 3.5 5.2

9 0

0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0. 0 . 0 . 0 . 0 . 0 . 0. 0 . 0. 3.9 3.0 2.5 2.3 2. I 2.0 1.9 1.8 1.8 1 . 1 1 .1 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.7 1.1 1.1 1.8 1.9 1.9 2.0 2 .2 2 .4

LOO

0. 0. 0 . 0. 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0. 0 . 0. 0 . 0 . 0 . 0. 0. 0 . 0 . 0. 0 . 0. 0 . 0 . 0. 4.1 3.1 2 - 6 2.4 2.2 2.0 1.9 1.9 1.8 1.1 1.1 1.1 1-6 1.6 1.6 1.6 1.6 1.5 1.5 1.5 1-5 1 -5 1.5 1.5 1.6 1.6 1.6 1.6 1.6 1.7 1.1 1.8 1.8 1.9 2.0 2.2 2.5 3.0

110

0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 4. 1 3.1 2.6 2.4 2.2 2.0 1.9 1.8 1.8 1.7 1.1 1.6 1.6 1.6 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.6 1.6 1 . 1 1.1 1.8 2.0 2.2 2.1 3.9

140.0 0. 0. 0 . 0. 0 . 0 . 0 .

120

0 . 0. 0 . 0. 0. 0 . 0. 0. 0. 0 . 0. 0. 0. 0. 0 . 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0 . 0 . 0. 0. 0. 0. 0. 0. 0 . 0 . 0. 0. 4.0 3.0 2.6 2.3 2.1 2.0 1.9 1.8 1.1 1.6 1.6 1.6 1.5 1.5 1.5 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.3 1.3 1.3 1.4 1.4 1.4 1.4 1 .4 1.4 1.5 1.5 1.6 1.7 1.9 2.2 3.2

120.0 0. 0.

130

0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 3.8 2.9 2.4 2.2 2.0 1.8 1.7 1.6 1.5 1.5 1.5 I .4 1.4 1.4 1.3 1.3 1.3 1.3 1.3 1.2 1.2

140

0. 0 . 0 . 0 . 0. 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0. 0. 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0. 0 . 0 . 0 . 0. 0 . 0 . 0. 0. 0 . 0. 3.4 2.6 2.2 1.9 1.8 1.1 1.6 1.5 1.4 1.4 1.3 1.3 1.3 1.2 1.2 1.2 1.2 1.1 1.1 1.1 1.1 1. I 1.1 1.1 1.1 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1 .O 1.0 1.0 1.0

150

0. 0 . 0 . 0. 0. 0 . 0 . 0 . 0 . 0 . 0. 0. 0. 0 . 0. 0. 0. 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0. 0. 0 . 0 . 0 . 0. 0. 0 . 0 . 0. 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0. 0. 3.0 2.2 1.9 1.7 1.5 1.4 1 - 3 1.3 1.2 1.2 1.1 1.1 1.1 1.1 1.1 1.0 1.0 I .o 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.1 1.2 1.3 1.8

60.0 0. 0. 0 . 0.

160

0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0. 0. 0. 2.4 1.8 1.5 1.4 1.2 1.7 1.1 1. I 1.0 I .o 1.0 I .o 1.0

1.1 1.1 1.2 1.4 2.0

40.0 0. 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 .

1.0

110

0. 0 . 0 . 0. 0 . 0. 0 . 0 . 0 . 0 . 0. 0. 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0. 0. 0. 0 . 0 . 0. 0. 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 1.6 1.3 1.1 1.0 1.0 1.0 1.1 1.2 1 . 1

19.9 0. 0.

0. 0 . 0 . 0. 0. 0. 0. 0. 0 . 0. 0 .

n.

DELTA

0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0. 0. 4.0 1.4 1.1 1.0 1.0 1.0 1.1 1.2 1.4 2.1

19.9 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0. 0 . 0 . 0. 0. 0. 0 . 0. 0. 0. 0 . 0. 0 . 0. 0 . 0. 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0. 0. 0 . 0. 0. 0. 0. 0 . 0 . 0 . 0. 0 . 0. 0. 0 . 0 . 0. 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0.

0. 0 . 0 . 0 . 0. 0. 0 . 0 . 0 . 4.6 1.6 1.1 1.1 1.1 1.0 1.0 1.0 1 .o I .o 1.0 1.1 1.1 1.1 1.2 1.3 1.4 1.6 2.0 3. I

39.9 0. 0 . 0 . 0 . 0 . 0. 0. 0 . 0. 0. 0. 0 . 0. 0. 0 . 0 . 0. 0. 0. 0 . 0. 0. 0. 0 . 0 . 0. 0. 0 . 0. 0. 0. 0 . 0. 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 .

1.0 1.0 1.0 1.0 1 .o 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.2 1.2 1.2 1.2 1.3 1.3 1.3 1.4 1.4 1.5 1.6 1.7 1.9 2.1 2.4 2.9 4.5

79.9 0. 0 . 0 . 0 . 0. 0 . 0. 0 . 0. 0 . 0. 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0. 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0. 0. 0 . 0. 0 . 0 . 0 . 0 . 0 . 0. 0 . 0. 0 . 0. 0. 0 . 0 . 0 .

0. 0. 0. 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 1.8 0. 0. 1.5 0. 0. 1.4 0. 0 . 1.3 0. 0. 1.3 119.5 0 .

2 .0 4.0 6.0 8.0

10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0 30.0 32.0 34.0 36.0 38.0 40.0 42.0 44.0 46.0 48.0 50.0 52.0 54.0 56.0 58.0 60.0 62.0 64.0 66.0 68.0 10.0 12.0 74.0 16.0 18.0 80.0 82.0 84.0 8 6 . 0 88.0 90.0 92.0 94.0 96.0 9 8 . 0

100.0 102.0 104.0 106.0

I . 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.3 1 - 3 1.3 1.3 1.3 1.4 1.4 1.4 1.5 1.5 1.5 1.6 1.1 1.8 1.9 2. I 2.3 2.6 3.2 4.9

99.9 0. 0. 0. 0. 0 . 0. 0 . 0 . 0 . 0. 0 . 0. 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0. 0 . 0 . 0. 0. 0 . 0. 0 . 0. 0. 0 . 0 . 0. 0 . 0. 0 . 0. 0 . 0 . 0 . 0. 0. 0 .

2.5 0. 2.0 0. 1.8 0. 1.6 0. 1.5 138.9

. 1.5 1.4 1.4 1.4 1.4 1.4 1.4 1.3 1.3 1.3 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.5 1.4 1.5 1.5 1.5 1.6 1.6 1.7 1.7 1.8 1.9 2.0 2.2 2.4 2.8 3.4 5.2

119.9 0. 0. 0. 0.

3.1 2.4 2.1 1.9 1.8 1 . 1 1.6 1.6 1.6 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.6 1.6 1.6 1.6 1.7 1.1 1.8 1.9 2.0 2. I 2.3 2.5 2.8 3.5

108.0 110.0 112.0 114.0 116.0 118.0 120.0 122.0 124.0 126.0 128.0 130.0 132.0 1 3 4 . 0 136.0 138.0 140.0 142.0 144.0 146.0 148.0 150.0 152.0 154.0 156.0 158.0 160.0 162.0 164.0 166.0 168.0 110.0 172.0 174.0 116.0 178.0

.. 5.3 0. 139.9 0. 0. 0. 0 . 0. 0. 0. 0. 0 . 0. 159.8 0. 0. 0. 0. 0. 0. 2.1 0. 0. 0. 3.3 0. 0. 0. 5.0 0. 0. 0. 117.6 0. 0. 0 . 0 . 0 . 0. 0 . 0 . .2

.2

.2

.2

.2

.2 - 2 - 2 - 2 .2 .2 .2 .2 .3 . 3

0 . 0 . 0 . 0 . 0. 0. 0. 0. 4.5 0. 0. 0. 0. 159.9 0. 0. 0. 0. 0. 0. 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0. 0. 0 . 0. 0. 0 . 0. 0. 0. 0 . 0. 0.

0 . 0. 0. 0. 0. 0 . 0 . 0 .

0 . 0. 0 . 0 . 0. 0. 0 . 0.

0. 0 . 0 . 0 . 0 . 0 . 0 . 0 .

0. 0 . 0 . 0 . 0. 0 . 0. 0. 1 . 4

I .6 .. .. 0 . 0 . 0 . 0 . 0 . 0 . 0 . . . . 0. 0 . 0. 0 . 0. 0. 0. 2.2 0. 0. 0. 0. 0. 0. 0 . 1 0 0 . 0 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 .

99

OELTA

2.0 4.0 6.0 8.0

10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0 30.0 32.0 34.0 36.0 38.0 40.0 42.0 44.0 46.0 48.0 50.0 52.0 54.0 56.0 58.0 60.0 62.0 64.0 66.0 68.0 70.0 72.0 14.0 16.0 78.0 80.0 82.0 84.0 86.0 88.0 90.0 92.0 94.0 96.0 98.0

100.0 102.0 104.0 106.0 108.0 110.0 112.0 114.0 116.0 118.0 120.0 122.0 124.0 126.0 128.0 130.0 132.0 134.0 136.0 138.0 140.0 142.0 144.0 146.0 148.0 150.0 152.0 154.0 156.0 158.0 160.0 112.0 164.0 166.0 168.0 170.0 172.0 174.0 176.0 178.0

10

0. 0 . 0. 0. 0. 0. 0. 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 1-4 1.1 1.0 1.0 1.0 1.1 1.2 1.4 2.0

19.9 0. 0. 0. 0 . 0 . 0 . 0 . 0. 0 . 0 . 0. 0 . 0. 0 . 0. 0. 0 . 0. 0. 0. 0 . 0. 0. 0. 0 . 0. 0. 0. 0 . 0 . 0. 0. 0. 0. 0 . 0 . 0 . 0 . 0 . 0. 0 . 0. 0 . 0. 0 . 0 . 0 . 0. 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 .

E T A * 50.0 BETA

20

0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 5.1 1.7 1.4 1.1 1.1 1.1 1.0 1.0 1.0 1.0 1.0 1.0 1.1 1.1 1.2 1.1 1.4 1.6 1.9 2.9

39.9 0. 0. 0. 0 . 0. 0. 0. 0 . 0. 0. 0. 0 . 0. 0. 0. 0 . 0. 0. 0 . 0 . 0 . 0 . 0. 0 . 0. 0 . 0 . 0 . 0 . 0 . 0. 0 . 0. 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 .

30

0. 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 5. I 1.8 1.4 1.2 1.2 1.1 1.1 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.1 1.1 1.1 1.2 1.2 1.2 1.3 1.4 1.5 1.7 1.9 2.4 1.6

19 .9 0. 0. 0. 0. 0 . 0. 0 . 0. 0. 0 . 0 . 0. 0. 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0. 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 .

40

0. 0. 0. 0. 4.2 1.5 1.3 1.2 1.1 1.1 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1 .o 1.0 1.0 1.0 1.0 1.1 1.1 1.1 1.1 1.1 1. I 1.2 1.2 1.2 1.3 1.3 1.4 1.4 1.5 1.6 1.7 1.9 2.2 2.7 4.1

79.9 0. 0. 0. 0.

50

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1. I 1.1 1.1 1.1 1. I 1.1 1.1 1. I 1. I 1.1 1.1 1.2 1.2 1.2 1.2 1.2 1.3 1.3 1.3 1.4 1.4 1.5 1.5 1.6 1.7 1.9 2.1 2. k 2.9 4.5 .~ ~~

0. 99.9 0. 0. 0. 0. 0. 0 . 0. 0 .

60

0. 0. 0 . 0. 0. 1.8 1.5 1.3 1.3 1.2 1.2 1.2 1.2 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.3 1.3 1.3 1.3 1.4 1.4 1.4 1.5 1.5 1.6 1.6 1.7 1.8 2.0 2.2 2.5 3.1 4. 6

7 0 80

0 . 0. 0 . 0. 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0 . 0. 0. 0. \o. 7.0 0. 2.4 0. 1.9 0 . 1.7 0. 1.5 0. 1.5 158.2 1.4 2.9

2.3 1.4 1.3 1.3 1.3 1.3 1.3 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.4 1.4 1.4 1.4 1.5 1.5 1.5 1.6 1.T 1.8 1.9 2.0 0. 0. 119.9

0. 0. 0. 2.2 0. 0. 0. 2.5 0. 0. 0. 3.1 0. 0. 0. 4.7 0. 0. U. 1 3 Y . Y 0. 0 . 0. 0 . 0 . 0. 0 . 0. 0 . 0 . 0. 0.

2.0 1.8 1.7 1.6 1.5 1.5 1.4 1.4 1.4 1.4 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 I .3 1.3 1.4 1.4 1.4 1.4 1.4 1.5 1 .5 1.5 1.6 1.7 1.7 1.8 2.0 2.2 2.5 3.0 4 -5 0. 0 . 0. 0.

0 . 0. 0. 0. 159.9 0. 0. 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 .

90

0 . 0. 0. 0 . 0. 0. 0 . 0. 0 . 0 . 0 . 0. 0. 0. 0 . 0 . 0 . 0. 0 . 0. 3.3 2.5 2.2 1.9 1.8 1.7 1.6 1.6 1.5 1.5 1.5 1.4 1.4 1.4 1.4 1.4 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.4 1.4 1 .* 1.4 1-k 1.5 1.5 1.5 1.6 1.7 1.8 1.9 2.0 2.3 2 .1 4. 2

0 . 0. 0. 0. 0. 178.3 0. 0. 0. 0. 0. 0 . 0 . 0. 0. 0. 0. 0. 2 .1 0. 0. 0. 0. 0. 0. 2.5 0. 0. 0. 0. 0. 0. 3.7 0. 0. 0. 0. 0 . 0 . 160.0 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 .

100

0. 0. 0 . 0. 0. 0. 0. 0 . 0 . 0. 0 . 0. 0. 0. 0. 0 . 0 . 0. 0. 0. 0. 0. 0. 0. 0. 3.5 2.7 2.3 2.1 1.V 1.1 1.T 1.6 1.6 1.1 1.5 1.5 1.4 1.k 1.4 1.4 1.4 1.1 1.3 1.1 1.3 1.3 1.1 1.1 1.1 1.1 1.1 1.1 1.3 1.3 1.3 1.1 1.1 1.1 1.3 1.1 1.1 1.4 1.4 1.4 1.4 1.5 1.5 1.6 1.7 1.9

0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 .

0 . 0. 0. 0 . 0 . 0. 0 . 0 . 0 . 0. 0 .

0. 0 . 0. 0. 0 . 0. 0. 0. 0 . 0 . 0 . 0 . 0. 0 . 0 . 0. 0. 0. 0 . 0 . 0. 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 .

110

0 . 0. 0. 0. 0 . 0. 0 . 0 . 0 . 0. 0. 0. 0 . 0. 0 . 0. 0 . 0. 0 . 0. 0 . 0 . 0 . 0 . 0 . 0. 0. 0. 0. 0. 3.6 2.8 2.3 2.1 1.9 1.a 1.7 1.6 1-6 1.5 1.5 1.4 1.4 1.4 1.4 1.3 1.3 1.1 1.3 1.3 1.3 1.3 1.1 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.3 1.3 1.3 1.3 1.3 1.3 1.4 1.4 1.5 1.6 1.8 2.1

0 . 0 . 3 . 0 0 . 0 . 140.0 0. 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0. 0. 0. 120.0 0. 0. 0. 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0.

120

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0 . 0 . 0 . 0 . 0 . 0 . 0. 0. 0 . 0. 0. 0 . 0 . 0. 0. 0 . 0. 0. 0. 0. 0. 0 . 0. 0 . 0. 3.0 2 .3 1.9 1.7 1.6 1.5 1.4 1.4 1.3 1.3 1.2 1.2 1.2 1.2 1.2 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1. I 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1 .o 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.1 1.1 1.1 1.2 1.3

70

0. 0 . 0. 0 . 0 . 0 . 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0 . 0. 0. 8.6 2.9 2 .2 1.9 1.7 1.6 1.5 1.4 1.4 1.3 1.3 1.2 1.2 1.2 1.2 1.2 1.2 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.0 1 .o 1.0 1.0 1.0 1 .o 1.0 1 .o 1.0 1.0 1.0 1.0 1.0 1 .o 1.0 1 .o 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1 .o 1.0 1.0

80

0. 0. 0 . 0. 0 . 0. 0. 0. 0 . 0 . 0. 0. 0 . 0.

2.6 2.0 1.8 1.6 1.5 1.4 1.4 1.3 1.3 1.2 1.2 1.2 1.2 1.2 1.2 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 I . 1. I . I . 1. 1. I . I . 1. 1. 1. 1. I . I . I . 1.0 1 .o 1.0 1.0 1.0 1.0 1.0 1.1 1.1 1.1 1.1 1.1 1.1 1 . 1 1.1 1.1

7.8

90

0 . 0. 0 . 0. 0 . 0. 0. 0. 0. 6.5 2.2 1.8 1.6 1.4 1.4 1.3 1.3 1.2 1.2 1.2 1.2 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.2 1.2 1.2 1.2 1.2 1.3

110 120

1.0 0. 1.0 0 .

130

0 . 0 . 0 . 0 . 0. 0. 0. 0 . 0 .

99.7 2.0 1.6 1 . 4 1.3 1.2 1.1 1.1 1.1 1.1 I .o I .o 1 .o 1 .o 1 .o 1 .o 1.0 1.0 1 .o 1.0 1.0 1 .o 1 .o 1 .o I .o I .o 1 .o 1.0 1 .o 1 .o 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.2 1.2 1.2 1.3 1.3 1.6 1 . 4 1.5 1.6 1.8 2.1

140

0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0.

79.6 2. I 1.6 1 .4 1.3 1.2 1.2 1.1 1.1 1.1 1.0 1.0 1.0 1 .o I .o 1.0 1.0 1.0 1 .o 1.0 1 .o 1.0 1.0 1.0 1.0 1.1 1.1 1.1 1.1 1.2 1.2 1.2 1.3 1.3 1.4 1.5 1.7 1.9 2.4 3.6

80.0 0 . 0 . 0. 0. 0. 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0. 0. 0. 0. 0 . 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. n.

150

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0. 0 . 0. 0. 4.7 1.7 1.4 1.3 1.2 1.2 1.1 1.1 1.1 1.1 1.1 I . I 1.1 1.1 1.1 1.1 1.1 I . I 1.1 1.1 1.1 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1

I60

0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0. 0. 0 . 0. 0. 1.9 1.5 1.3 1.2 I . I 1 .o I .o 1 .o 1.0 1 .o I .o 1.0 1.1 1.1 1.2 1.3 1.5 1.0 2.7

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i70

0 . 0 . 0. 0. 0 . 0 . 0 . 0. 0. 0 . 0. 0. 0 . 0 . 0 . 0. 0 . 0. 0. 0 . 0. 0 . 0. 0 . 0. 0. 0 . 0. 0. 0. 1.5 1.2 1.1 1.0 1.0 1.0 1.1 1.3 2.0

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n.

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n. n. 0. 0. 0. 0. 0. 0. n.

2 . 0 4.0 6.0 8.0

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n.

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I .o 1.0 1.0 1 .o I .o 1.0 1.0 1.0 1.0 I .O 1 .o I .o 1.0 1.0 1.0 1.0 1.0

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1.2 7 .8 0. 0. 0. 0 . 1.2 179.2 0. 0. 0 . 0. 1.2 0. 0 . 0 . 0. 0.

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106

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30

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40

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80.0 0. 0. 0. 0.

50

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1.8 1.7 1.6 1.5 1 . 4 1.4 1.3 1.3 1.2 1.2 1.2 1.2 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.1 I . 1 1.1 1.2

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70

0. 0. 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0 . 0. 0. 0 . 0. 3.2 2.5 2.1 1.9 1.7 1.6 1.5 1.5 1.4 1.4 1.4 1.3 1.3 1.3 1.3 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1

80

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 9.2 3.0 ?. 3 2.0 1.8 1.7 1.0 1.5 1.5 1.4 1.4 1.3 1.3 1.3 1.3 1.3 1.2 1.7 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.2 1.7 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.3 1.3 1.3 1.4 1.4 1.5

90 100 110 120

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1 .o 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.1 1.1 1.1 1.1 I. L 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.2 1.2 1.2 1.2 1.2 1.3 1.3 1.3 1.3 1.4 1.4 1.5 1.6 1.7

130

0. 0. 0. 0. 99.8 1.6 1.3 1.2 1.1 1.1 1.1 1.0 1.0 I .O 1 .o 1.0 1.0

140

0. 0. 0. 0. 0. 0. 0. 0. 0. I9.7 1.9 1.5 1.3 1.2 1.1 1.1 1.1 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.1 1.1 1.1 1.1 1.1 1.2 1.2 1.2 1.3 1.3 1.4 1.5 1.6 1.8 2.1 2.5 3.8 79.9 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

150

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 59.7 2.0 1.5 1-3 1.2 1.1 1.1 1.1 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1 .o 1 .O 1.0 1.1 1.1 1.1 1.2 1.2 1.3 1.3 1.5 1.6 1.8 2.3 3.4

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160

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19.9 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

0. 0. 0. 0. 0. 0.

n.

109

8

E T A - 150.0 B E T I

D E L T A

2.0 4.0 6.0 8.0

10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0 28 .0 30.0 32.0 34.0 36.0 38.0 40.0 42 -0 44.0 46.0 48.0 50.0 52.0 54.0 56.0 58.0 60.0 62.0 64.0 66.0 68.0 70.0 72.0 74.0 76.0 78.0 80.0 0 2 . 0 84.0 86.0 88.0 90.0 92.0 94.0 96.0 98.0

100.0 102.0 104.0 106.0 108.0 110.0 112.0 114.0 116.0 118.0 120.0 122.0 124.0 126.0 128.0 130.0 132.0 134.0 136.0 138.0 140.0 142.0 144.0 146.0 148.0 150.0 152.0 154.0 156.0 158.0 160.0 162.0 164.0 166.0 168.0 170.0 172.0

10

0. 0 . 0. 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0 . 0. 0. 0 . 0 . 0. 0. 0. 0. 0. 0. 0. 0 . 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0. 1.7 1.3 1 .1 1.0 1.0 1.0 1.1 1.2 1.6

20.0 0. 0. 0. 0. 0 . 0. 0. 0 .

2 0

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0 . 0 . 0. 0. 0. 0 . 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0. 2.5 1.9 1.6 1.4 1.3 1.2 1.2 1.1

2: 1.0 1 .o 1.0 1.0 1.0 1.0 1.1 1.2 1.7

40.0 0. 0. 0.

174.0 176.0 118.0 0. 0.

3 0

0. 0. 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0 . 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0 . 0 . 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0. 0. 3.2 2.4 2.1 1.8 1.7 1.5 1.5 1.4 1.3 1.3 1.2 1.2 1.2 1.1 1.1 1.1 1.1 1.1 1.1 1 .o 1 .o 1.0 1.0 1.0 1 .o 1.0 1.0 1.0

40

0. 0. 0. 0. 0. 0. 0 . 0 . 0. 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 3.8 2.9 2.4 2.2 2.0 1.8 1.7 1.6 1.6 1.5 1.5 1 .4 1.4 1.4 1.4 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3

50

0. 0. 0 . 0 . 0. 0 . 0 . 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0 . 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0 . 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 4.3 3.2 2.7 2.4 2.2 2.1 2.0 1.9 1.8 1 . 7 1.7 1.6 1.6 1.6 1.6 1.6 1.5 1.5 1.5

60

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0. 0. 5.6 3.5 2.9 2.6 2.4 2.3 2.1 2.0 2.0 1.9 1.9 1.8 1.8 1.8 1.7 1.7 1.7 1.7 1.7 1.8 1.8 1.8 1.9 1.9

70

0 . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0 . 0 . 0. 0 . 0. 0. 0 . 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0 . 0 . 0. 0. 0 . 0. 0 . 0. 4.8

80

0. 0. 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0 . 0. 0. 4.8 3.6 3 .1 2 . 8 2.6 2.5

90 100 110 120

0 . 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0. , 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0 . 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0. 0 . 0. 0. 0. 0. 119.0 0. 0. 0. 3.4 0 . 0. 0. 2.6 0. 0. 0. 2.3 0. 0. 0. 2.1 0. 0. 138.1 2.0 0. 0. 3.9 1.9 0. 0. 3.0 1.8 0. 0. 2.6 1.8 0. 0. 2.4 1.8 0. 0. 2.2 1.1 0. 6.4 2.1 1.7 0. 3.4 2.0 1.7 0. 2.9 2.0 1.7 0. 2.6 1.9 1.7 0. 2.4 1.9 1.8 4.6 2.3 1.9 1.8 3.6 2.2 1.9 1.8 3.0 2.1 1.9 1.8 2.7 2.1 1.9 1.9 2.5 2.0 1.9 1.9 2.4 2.0 1.9 2.0 2.3 2.0 1.9 2.1 2.2 2.0 2.0 2.2 2.1 2.0 2.0 2.3 2.1 2.0 2.1 2.5 2.1 2.0 2.1 2.8

3.6 2.3 2.0 2.0 2.2 3 - 2 3.1 2.2 2.0 2.0 2.3 3.9 2.7 2.1 2.0 2.1 2.4 5.9 2 .5 2 - 1 2.0 2.1 2.6 119.8 2.4 2.0 2.0 2.2 2.9 0. 2.3 2.0 2.0 2.8 3.3 0. 2.2 2.1 2.0 2.0 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.Y 1.9 2.0

2.0 2.0 2.0 2.1 .~ i.0 2.1 2.0 2.2 2.0 2.2 2.0 2.3 2.0 2.5 2 - 0 2.6 2.1 2.9 2.2 3.3 2.2 4.0 2.4 6.0 2.5 176.4 2.7 0.

2.4 4 . 1 2.5 6.1 2.7 139.8 2.9 0. 3.3 0. 4.1 0. 6 . 1 0.

159.b 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0 .

2.0 3.1 0. 0. 0. .. ~. 2.1 3.7 0. 0. 0. 2 . 2 5 .6 0. 0. 0. 2.3 159.7 0. 0 . 0. 2 .5 0. 0. 0. 0. 2.8 0. 0. 0. 0 . 3 . 4 0. 0 . 0. 0.

0. 0. 0. 0. 0. 0. 0. 0. 0 . 0 . 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0.

130 140 I50 160 170

0. 0. 1.0 0. 0. 0. 0. 1.0 0. 0. 0. 0. 1.0 0. 0. 0. 0. 1.0 0. 0. 0. 7 9 - 8 1.0 39.8 0. 0. 1.9 1.0 1.4 0. 0. 1.6 1.0 1.2 0. 0. 1.5 1.0 1.1 0. 0. 1.4 1.0 1.0 0.

99.5 1.3 1.1 1.0 19.b 2.7 1.3 1.1 1.0 1.3 2 . 1 1.3 1.1 1.0 1.1 1.9 1.3 1.1 1.0 1.0 1.8 1.3 1.1 1.0 1.0 1.7 1.3 1.1 1.1 1.0 1.6 1.3 1.2 1.1 1.1 1.6 1.3 1.2 1.1 1.7 1.6 1.3 1.2 1.2 1.4 1.5 1.3 1.3 1.3 2.1 1.5 1.4 1.3 1.4 19.8 1.5 1.4 1.4 1.5 0. 1.5 1.4 1.4 1.7 0. 1.5 1.5 1.5 2.1 0. 1.6 1.5 1.6 3.2 0. 1.6 1.5 1.7 39.9 0. 1.6 1.6 1.9 0. 0. 1.6 1.7 2.2 0 . 0. 1.7 1.8 2.7 0. 0. 1.7 1.9 4.2 0. 0. 1.8 2.1 59.9 0. 0. 1.8 2.3 0. 0. 0. 1.9 2.6 0. 0. 0. 2.0 3.2 0. 0. 0. 2.1 4.9 0. 0. 0. 2.3 79.9 0. 0. 0. 2 . 6 0 . 0. 0. 0. 2.9 0. 0. 0. 0. 3.6 0. 0. 0. 0. 5 . 5 0. 0. 0. 0.

99 .9 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0 . 0 . 0. 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0 . 0 . 0. 0 . 0 . 0. 0. 0. 0. 0. 0 . 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0 . 0. 0. 0 . 0 . 0 . 0. 0. 0 . 0. 0 . 0 . 0. 0. 0. 0 . 0 . 0. 0 . 0. 0. 0 . 0 . 0 . 0 . 0. 0. 0. 0 . 0. 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0. 0 . 0. 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0. 0 . 0. 0 . 0. 0 . 0 . 0. 0. 0 . 0. 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0 . 0. 0 . 0 . 0. 0. 0. 0 . 0.

5 . 0 0 . 0. 0. 0. 0. 0. 0. 0. 0. n. 1.5 2.0 139.9 0. 0. 0. 0. 0. 0. 0 . 0. 0. 0. 1 . 5 2 . 2 0 . 0. 0 . 0. 0. 0. 0. 0. 0. 0. 0. 1.6 2.4 0. 0. 0. 0. 0. 0- 0. 0 . 0. 0. 0. 1.6 2.9 0. 0. 0. 0. 0 . 0 . 0- 0- 0. 0. 0. 1.6 4.3 0. 0. 0 . 0. 0. 0. 0- 0. 0 . 0. 0. 1.7 120.0 0. 0 . 0 . 0 . 0. 0. 0 . 0 . 0. 0. 0- 1 . 8 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 2.0 0. 2.4 0.

1.3 3.4 0. 1.3 100.0 0. 1.4 0. 0. 1.5 0. 0. 1.7 0. 0. 2.3 0. 0.

80.0 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0 . 0.

0. 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0.

.. ._ 0. 0. 0. 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0 . 0 . 0. 0. 0 . 0 . 0. 0 . 0. 0. 0. 0. 0. 0 . 0. n. 0.

0. 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0 . 0 . 0. 0. 0. 0 . 0. 0 . 0. 0. 0 . 0. 0. 0. 0 . 0. 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0.

0. 0 . 0. 0. 0 . 0. 0. 0. 0 . 0 . 0. 0.

0 . 0. 0. 0. 0 . 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0, 0. 0. 0. 0 . 0.

1 . 0 0 . 0 . 0. 0. 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 .

110

.

.

D E L T A

2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0 30.0 32.0 34.0 36.0 38.0 40.0 42.0 44.0 46.0 48.0 50.0 52.0 54.0 56.0 58.0 60.0 62.0 64.0 66.0 68.0 70.0 72.0 74.0 16.0 70.0 80.0 82.0 84.0 86.0 88.0 90.0 92.0 94.0 96.0 98.0

100.0 102.0 101.0 106.0 108.0 110.0 112.0 114.0 116.0 118.0 120.0 122.0 124.0 126.0 128.0 130.0 132.0 1 3 4 . 0 136.0 138.0 140.0 142.0 144.0 146.0 148.0 150.0 152.0 154.0 156.0 158.0 160.0 1 6 2 . 0 164.0 166.0 168.0 170.0 172.0 174.0 176.0 178.0

0. 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1.8 1.4 1.2 1.1 1 .o 1 .o 1.0 1.1 1.5

20.0 0. 0. 0. 0.

20

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 2.8 2.1 1.8 1.6 1.4 1.4 1.3 1.2 1.2 1.1 1.1 1.1 1.1 1 .O 1.0 1.0 1.0 1 .O 1.0

30

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 3.7 2.8 2.4 2.1 1.9 1.8 1.7 1.6 1.6 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.6 1.8 2.5

60.0 0. 0. 0. 0.

40

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 4.4 3.4 2.9 2.6 2.3 2.2 2.1 2.0 2.0 1.9 1.9 1.9 1.9 1.9

50

0. 0. 0. 0. 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 5.1 3.9 3.3 2.9 2.7 2 . 6 2.4 2.4 2.3 2.3 2.2 2.2 2.3

60

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 5.5 4.2 3.6 3.2 3.0 2.8 2.1 2.6 2.6 2.5 2.5 2.5

70 80

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 5.9 0. 4.6 0. 3.9 0. 3.5 0. 3 . 3 5.8 3.1 4.5 3.0 3.8 2.9 3.4 2.9 3.2 2.9 3.0 2.9 2.9 2.9 2.8 3.0 2.8 3.1 2.7 3.3 1 2.8 3.6 2.8 4.0 2.8 4.0 2.9 7.2 3.1 159.3 3.3 0. 3.1 0.

PO

0. 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 5.9 4.5 3.9 3.5 3 . 3 3.1 3.0 3.0 2.9 2.9 2.9 3.0 3. I 3 .2 3 .4 3.7 4.2 5.0 7.5

74.3 0. 0. 0 . 0. 0. 0. 0.

LOO 110 120

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. I). 0. 0. 0. 0. 0. 0. 0. 0. 0. 0 . 4.6 0. 0 . 3.6 0. 0 . 3.1 0. 0 . 2.9 0. 0. 2.7 0. 5.2 2.6 0. 4.0 2.6 0. 3.5 2.5 0. 3.2 2.5 0. 3.0 2.6 5.6 2.9 2.6 4 . 3 2.8 2.1 3.8 2.8 2.8 3.4 2.7 2.9 3.2 2.8 3.1 3.1 2.8 3.4 3.0 2.9 3.9 2.9 3.0 4.7 2.9 3.1 7.1 2.9 3.3 119.7 2.9 3.6 0. 3.0 4.1 0- 3.1 5.0 0. 3.2 7.5 0. 3 . 4 1 3 9 . 5 0. 3 . 7 0 . 0 . 4 .2 0. 0. 5.1 0. 0. 7.6 0. 0.

159.1 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

130

0. 0. 0. 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0. 0. 98.8 3.9 3.1 2.7 2.5 2 . 4 2.3 2.3 2.2 2.2 2.3 2 . 3 2.4 2.5 2.6 2.8 3.1 3.5 4 . 3 6.5

99.7 0. 0. 0. 0. .o . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

140

0. 0. 0. 0. 0. 0. 0. 0. 0.

79.3 3 . 0 2.4 2 .2 2.0 1.9 1.9 1.9 1.9 1.9 1.9 2.0 2.1 2.1 2.3 2 . 4 2.1 3.1 3.8 5.7 19.8 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

150

0. 0. 0. 0. 59.7 2.0 1.1 1.6 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.6 1.7 1.1 1.9 2.0 2.2 2.5 3.1 4.7 59.8 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

160

1.0 1 .o 1.0 1.0 1.0 1.0 1.1 1.1 1.1 1.2 1.2 1.2 1.3 1.4 1.5 1.7 1.9 2.4 3.6

39.8 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

170

0. 0. 0. 0. 19.7 1 .2 1.1 1.0 1.0 1.0 1.1 1.3 1.5 2.3 19.8 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

2 . 6 4 . 5 0 . 0. 0. 0. 0- 0. 0. 0. 0. 0. 2.7 6.6 0. 0. 0. 0. 0. 0. 0. 0. 0 . 0. 2 . 8 1 3 9 . 7 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0. 3.00. 0. 0. n. 0. 0. 0. 0. 0. 0. 0. 3 . 3 0 . 0 . 0. 0. 0. 0. 0. 0 . 0 . 0 . 0 . 4 . 0 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 5 . 9 0 . 0 . 0. 0. 0. 0. 0. 0 . 0 . 0. 0. 2.3 .

2.4119.80. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 2.6 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 2 . 0 0 . 0 . 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0. 3 . 4 0 . 0 . 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0. 4 . 9 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

2.0 99.9 0. 0. 0. 2.1 0. 0. 0. 0. 2.3 0. 0. 0. 0. 2.6 0. 0. 0. 0- 3.8 0 . 0. 0. 0.

00.0 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0 . 0 . 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

0. 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

111

ElA.17O.O BETA

O E L T I

2.0 4.0 6.0 8.0

10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0 30.0 32.0 34.0 36.0 38.0 40.0 42.0 44.0 46.0 48.0 50.0 52.0 54.0 56.0 58.0 60.0 62.0 64.0 66.0 68.0 70.0 72.0 74.0 76.0 78.0 80.0 82.0 84.0 86.0 88.0 90.0 92.0 94.0 96.0 98.0

100.0 102.0 104.0 106.0 108.0 110.0 112.0 114.0 116.0 118.0 120.0 122.0 124.0 126.0 128.0 130.0 132.0 134.0 136.0 138.0 140.0 142.0 144.0 146.0 148.0 150.0 152.0 154.0 156.0 158.0 160.0 162.0 164.0 166.0 168.0 110.0 172.0 174.0 116.0 11n.o

10

0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0. 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0. 0. 2.1 1.6 1.3 1.2 I . 1 1 . 1 1.0 1.0 1.0

2 0

0. 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0. 0 . 0. 0 . 0. 3.5 2.7 2.3 2.1 2.0 2.0 2.0 2.2 2.9

39.9 0. 0. 0.

30

0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0.

3.7 3.3 3.0 2.9 2.9 3.0 3.4 4.8

59.8 0. 0. 0. 0 . 0 . 0. 0 . 0 .

4.n

0 . 0 .

40

0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0. 0 . 0. 0 . 0 . 0. 0 . 0 . 0. 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0. 0. 0 . 0. 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 6 .0 4 . 7 4 . 1 3.8 3.7 3.7 4.0 4.5 6.4

79.7 0. 0. 0. 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 .

50 60 70 80 9 0 LOO 110 120 130 I40 150 160 170

0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0. 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0. 0 . 0. 0. 0. 7.0 5.5 4.8 4.5 4.4 I 4.5 4.8 5.5 7.9

99.5 0. 0. 0. 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 .

0. 0. 0 . 0 . 0. 0 . 0. 0. 0. 0 . 0 . 0 . 0 . 0 . 0. 0 . 0. 0 . 0. 0 . 0 . 0. 0. 0 . 0. 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0. 0. 0 . 0. 0 . 0 . 0. 0 . 0. 0 . 0 . 0. 0 . 0 . 0. 0 . 0. 0 . 0. 0. 0 . 0 . 0. 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0. 0 . 0. 0. 0 . 0 . 0. 0 . 0. 0 . 0. 0. 0 . 0 . 0. 0 . 0. 8 .4 0. 6.6 0. 5.8 0. 5.5 0. 5.4 7.8 5.5 6.1 6.0 5.4 7.0 5.1 5.0 5. I 5.5 6 . 3 9. I

19.2 0. 0. 0. 0 . 0. 0 . 0. 0 . 0 . 0 . 0 . 0. 0 . 0 . 0. 0 . 0. 0. 0 . 0 . 0 . 0 . 0. 0 .

10. I 138.1

0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0.

0. 0. 0. 0. 0. 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0. 0. 0 . 0 . 0. 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0. 0 . 0. 0 . 0 . 0 . 0 . 0 . 8.7 6.8 6. I 5.8 5.7 5.8 6.3 7.4

10. 1 157.3

0 . 0 . 0. 0. 0 . 0 . 0 . 0 . 0. 0. 0 . 0 . 0 . 0. 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0. 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 .

n.

0. 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0. 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0. 0. 0. 8.7 6.9 6. I 5.8 S.8 5.9 6.4 7.6

11.0 168.9

0. 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . ~. 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 .

0. 0 . 0. 0 . 0. 0 . 0. 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0. 0 . 0. 0. 0. 0 . 0. 0 . 0. 0 . 0 . 0 . 0. 0. 8.S b . 7 6.0 5.7 5.7 5.9 6.4 7.5

11.0 157.0

0. 0. 0. 0. 0. 0 . 0. 0. 0. 0 . 0 . 0 . 0. 0 . 0. 0. 0. 0 . 0. 0 . 0 . 0 . 0. 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 .

0 . 0. 0 . 0. 0 . 0. 0 . 0. 0 . 0. 0 . 0 . 0 . 0. 0 . 0. 0 . 0. 0 . 0. 0 . 0. 0 . 0. 0 . 0 . 0 . 0. 0 . 0. 0 . 0. 0 . 0 . 0 . 0 . 0 . 0. 0. 0. 0 . 0. 0 . 0. 0 . 0. 0. 0. 0 . 0 .

0 . 8 .0 6.4 5.7 5.5 5.4 1 5.6 6. I 7.3

10.6 138.4

0. 0. 0 . 0 . 0. 0 . 0 . 0. 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0. 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 .

5 - 0 5.2 ~ ~~

5.7 6.8 9.9

18.9 0. 0 . 0. 0. 0. 0. 0. 0 . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0. 0 . 0. 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0. 0 . 0 . 0. 0 . 0 . 0 .

0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0. 0. 6.3 5.1 4.6 4.4 4.4 4.6 5.1 6 . I 8.9

99.2 0. 0. 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 .

0 . 0 . 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0.

5. I 4.2 3.8 3.7 3 .1 3.9 4.3 5.2 7.7

79.3 0. 0. 0. 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0. 0 . 0 . 0 . 0 . 0.

70.1

0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 .

58.8 3.8 3.2 2.9 2.9 2.9 3.1 3.5 4.2 6.2

59.5 0. 0. 0. 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0.

0 . 0 . 0 . 0 .

39.4 2.4 2.1 2.0 2.0 2. I 2.2 2.5 3.0 4.5

39.6 0. 0. 0. 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 .

1.0 1.0 1.1 1. I 1.2 1.3 1.4 1.8 2.6

19.7 0. 0. 0. 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0. 0. 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0. 0 . 0 . 0. 0 . 0 . 0 . 0 .

0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 .

0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0 . 0. 0. 0 . 0 . 0 .

0 . 0 . 0 . 0 . n.

112

I

I I I

APPENDIX E

I I I I I

7

c

Removing Uncertainty in t o Caused by a Quantized Time Base in the Spacecraft

Assume that H .-- 0.

Assume that the time base is quantized in the spacecraft such that the time at which the digitized fan crosses the sun is known to lie in the time interval t o

between to, and to, . One revolution later the next time the fan crosses the sun the "see sun" time is t, which lies in the interval between t,, and t 1B . See Figure A.

W W v)

2 3 v)

W W v)

W W v)

z 3 v)

W W v)

Figure A-Sun sightings defined within a quantized time base

Since e = 0 the spin period T is constant and is stable over a long period of time. Thus T may be calculated very accurately over a large number of revolutions.

Now from Figure A we can see that

where

I We can also write

tmin L (t ,* - 3T) t - < ( t3B - 3T) ma x

e t c . c t c .

It should be possible to find a tmin and a tmax which will satisfy all the above inequalities. If the inequalities are satisfied over a sufficiently large number of revolutions, the value of tmin should approach that of t m a x . Thus it is possible to define the value of to within any desired accuracy by merely solving the inequalities over a sufficiently large number of revoltuions.

This method of solution is vulnerable to noise. It is therefore advisable to test to make sure that tmin 5 t holds at all times. max

114

c

i

b

spAcEcz?

APPENDIX F

Horizon sensors which operate in the visible spectrum may be used a s long as the earth as seen from the satellite is fully illuminated. From Figure AA it can be seen that the spacecraft sees a fully illuminated earth as long as

77 t (90" - p) < 90"

or

TO +--

Figure AA-Relationships of spacecraft to sunlit earth

c

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(CATEGORY) ATTITUDE DETERMINATION OF SPIN STABILIZED ... · X-631-65-13 ATTITUDE DETERMINATION OF SPIN STABILIZED SPACECRAFT USING A DIGITAL ASPECT SENSOR March 1965 Goddard Space