Determination of orbital elements and refraction

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00

~ME.MORANDUM REPORT NO. 1357

liii~ 11h

waIluh I

DETERM INATION OF ORB ITAL ELEMENTS AND REFRACTION EFFECTS FROM SINGLE PASS DOPPLER OBSERVATIONS

R. B. Patton, Jr. V. W. Richard

AS T 1A OCT? 0 19W

0

of the Army Project No. 503-06-011 ordnance Management Structutre Cu No. 5210.11. 143 BALLISTIC RESEARCH LABORATORIES TODqmr,m

ABERDEEN PROVING GROUND, MARYLAND

ASTIA AUIIABILrrY NOTICE Qualified requestors way obtain copies of this report fro

ASTIA..

This report vill be published in the proceedings of the Symposium on Space Research and thereby will be available to the public.

I

RESEARCH

BALLISTIC

LABORAT

RI

S

MEMORANDUM RPORT 710. 1357 JME 1961

OF ORBr"AL ELKWTS AND FRACT!-!

IP ATI EFFECTS FROM SINGIE PASS DOPPLER OBSERVATICKS

R. B. Patton, Jr. V. W. Richard

Ballistic Measurements LAboratory

Presented at the Symposium on Space Research, Florence, Italy, April 1961

jrdADR

twmb of the Army roJect No. 503-06-011 tructure Coe No. 201Y1

DwEment

ABIRDZIN

PROVING

A3

GROUND, MARYLAND

BALLItSTIC

L13ORLT-OEIS

RESEARCH

30. 1357'

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LODL TT. A method has been deyelo ed for th" determination of a ccsletc set of orbital parameters from a few minutes of Doppler data recorded in the course of a single pass of a satellite.

may be

The source of the sinal

a transmitter in the satellite or a ground-based transmitter reflecting a signal fro

the satellite.

The latter transmitting system requires

more costly and complex equipment but offers reliability, an accurately known transmitter frequency, and a stronger ge

try for a more accurate

orbital computation when the number of receiving stations is Since it

limited.

was desired to develop a rapid, reliable, and moderately

accurate method of determining the orbital parameters of a satellite tracked by a Doppler system eploying a m4inm

of receiving stations,

ephasis was placed on the development of a solution from single lass observations recorded at fron

e to three receiving sites.

The single

pass limitation vas conslazd to present a challenging and worthy problem for which there would be numerous mpplications if

a reasonable

solution could be developed. in the past, Doppler data have been used primarily to measure the slope and time of inflection of the frequency-time curve to obtain slant

range and time-of-closest-approach information.

This is considered to

be only an elemntal use of the information in the Doppler data.

eingle

pass observations from one receiver have been demonstrated to contain sufficient information for satellite orbital determinations of suffl'

-it

accuracy for many applications.

For exaa ple, it

may be desirable to know the orbital parameters as

quickly as possible after launching a satellite.

The orbital parameters

of a newly launched satellite could be computed within beginning of its

free flight.

Again,

i-u"

after the

after attempting to deflect or

steer a satellite into a different orbit, it

may be desirable to kncY

the new orbital pwaieters within a matter of minutes.

The fllcwing sections will digcusa terminations from DoppIle

4

'e

practicality of orbit cie-

data alone end will indicate limitations as

well as the obvious advantages for zhis conceptually simple technique.

5

DESCRIICN OF TRACNG EQU~eNT AND DATA Doppler observations cnu a function of time.

ist of recordings of Doppler frequency, as

Here the Doppler frequency is defined to be the fre-

quency obtained by heterod~yning a locally generated signal against the signal received from the satellite followed by a correction for the frequency bias introduced as a result of the difference between the frequency of the local oscillator and that of the signal source. the Doppler frequency is

3h tais report,

defined to be negative when the satellite is

approaching the receiving site and positive when it

is

receding.

If

the

Doppler frequency, as defined, is plotted as a function of time, one obtains a curve of the form shown in Figure 1, usually referred to as an IS" -curve.

The aoymnietry of the curve is typical for a tracking system with a ground-based transmitter and a receiver separated by an appreciable distance.

Oly for a satellite whose orbital plane bitecta the base line

will the Doppler data produce a symtrical system.

"S" curve with a reflection the "8" curve is very nearly

With a satellite-borne tranmitter,

symmetrical,

being modified slightly by the Earth's rotatin and the

refractive effect of the ionosphere.

If

continuous observations are made

such as ore per second, Figure 1 (a)

and sampled at frequent intervals,

illustrates an analog plot of the data avaiL. le for ccmputer input.

However,

with a ground based transmitter it

number of observations in For example, it

may be necessary to limit the

order to minimize equipnt

cost and coplexity.

is possible to use only three antenna beams and provide

three sections of the "S" curve as shown in Figure 1 (b). bility

is

Another possi-

the use of a scanning antenna beam to provide discreet observations

at regular intervals as shown in Figure 1 (c).

Such data could be obtained

by an antenna with a thin, fan-shaped beam which scanned the sky repetitively.

The data in any of the forms suggsted above my be used readily as input for the computing proct-dure.

Whenever possible,

thl

input con-

sists of the +otal 1N-ppler cycle count over a variable time interval re:t than the Doppler frequency itself

(i.e. the arca *a2u:i

of curves presented in Figure 1 (a) and 1 (b)).

6

the curves or arcs

TIME

(a)

CIM

(c) Fig.I-Dople freuenc-tie

cuves 7E

In order to handle the Doppler date rapidly and mccurtely, the automatically counted and digitized at the receiving

DOppler frequency is sites.

Figure 2 shows a simplified block diagram of a DOPLOC receiving

syntem.

Automatic,

real-time counting rf the Doppler frequency requires

a signal of high quality, that is duced by noise.

one with very sm

errors intro-

rand-

Doppler data, which are essentially noise free, are made

possible in the DOPLOC system by use of a very narrw bandwidth, phase(ref. 1) following the receiver.

locked, trackiug filter

Significant

improvements in the signal-to-noise ratiL& of noisy received signal

the use of

realized by extreme reduction of the system bandwidth tlrugh

the filter. available. is

Bandwidths adjustable from 1 to 100 cycles per s-cond are The filter

is

capible of phase-locked operation when a signal

400).

a noise-to-sign-

(i.e.

an weak as 36 decibels below the noise,

ratio o

are

power

The filtering action is obtained by use of a frequency-

controlled oscillator that is

correlated or phase-locked to the input

The basic block diagram of the tracking servo loop is shown in Figure 3. Tracking is accomplished with an electronic servo system designed signal.

to force the frequency-controlled oscillator to follow the vriatiocns of Correlation is maintained with

frequency and phase of the input signal.

tie

respect to input signal phase, frequency, first

derivative of frequency,

and with a finite but smll phase error, the second tim derMytive of

frequency.

This is done by a cross-correlation detector cona:sting of the

phase detector and filter, or equalizer network. the control voltage to the oscillator is

Ubder dynoafÂŁc conditions,

so filtered in the equalizer net-

work that tracking faithfully reproduces the rate of change of the input frequency.

An inherent feature of this design is

an effective acceleration

memory wbich provides smooth tracking and extrapolation through signal dropouts.

Experience with signal reception fro

Earth satellites has barne

out the necessity for this amory feature, since the ree$V.id signal ampliworks through signal null

The filter

tude my vay widely and rapidly.

periods very erfectively without losing lnIr.

In addition, ti

s

.

provides effective tracking of the desired Doppler signl in the -'-esene are within receiving range

of interfering rignals when several sateili+s at the same time.

, ,

s

I

I

I

I

I

I

I8

00 z

h0

-zw

I w

_j a

oj0

tcc 4-C

0o-0

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w9

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N 'Al ICD Co 'Ii

IN

Ii.

0

w

U

4 I-.

U U, S 0 WICO

0

a

I0.

z

:0

The signal-to-noise power imprcirenent furnished by the tracking is

filter

equal to the ratio of tLe irput source noise beadwidth to

the filter

negligibly small at all bandwidths. output signal-to-noise is wiih

is

The internal noise generated by the filter

bandwidth.

The relation between input and

shown in Figures 4 and 5.

a receiver bandwidth of 10 kc and a filter

In a typical case

bandwidth of 5 cps,

signal buried 27 db down in the noise will appear at the filter

a

output

An experimental investigation

with a 6 db signal-to-noise rat.'o.

(ref. 2) has been made of the relation between signal-to-noise ratio and the uncertainty or random error in measuring the frequency of a Doppler signal.

The test results showing R.M.S. frequency error as a

function of signal-to-noise ratio and tracking filter shown in Figure 6.

bandwidths are

For the example cited previously, a signal 27 db

down In the noise can be read to an accuracy of 0.15 cps.

An integration

time or counting time interval of one second was used for these measurements.

The tracking filter tic lock-on system. frequencies,

can be equipped with a signal search and autcma-

Signals 30 db down in the noise at typical Doppler

from 2 to 14 kc,

second and the filter

can be detected within a fraction of a

phase-locked to the signal.

With this equipment,

signal acquisition and lock-,= have become routine in field operations.

The DOPLOC system has been used extensively for satellite tracking.

The inherent high sensitivity of the receiving system to signals of very 20 watts, low energy (2 x 10O

- 197 dbw, or 0.001 microvolts across gn ohms

for a threshold signal at 1 cps bandwidth) has permitted the use of conventional, low gain, wide coverage, antennas to achieve horizon to horizon tracking at great ranges.

It

has been found to be practical to chaige

bandwidths over the selectable range of 1 to 100 cps in accordance with the information content of the signal and thus achieve maximum signal-tonoise ratio.

Since the key to successful determination of orbits lies in

obtaining data with small values of random an. systematic error, t:t

quality dr.t

o,'? ut of the DOPLOC system has been an imjortao 4

high

eature.

An orbital solution, develuped specifica.,1y for this system, has yielded relatively accurate results with a surprisingly small number of DOPLOC tracking observations. i1

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INP OISE SOION TO 0 61

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THE ORBITAL SO.

-2CQ in

The method of solution consists of a curve-fitting procedure,

which a compatible set of approximations for the orbital jArameters, are improved by successive differential corrections.

The latter are

obtained from a least-squares treatment of an over-determined system of equations of condition.

The imposed limitation of single pass

detection permits several assumptinns which considerably simplify the computing procedure.

the assumption that the Earth

Among these is

may be treated dynamically as a sphere while geometricully regarding it

drag is

in accuracy will result if these assumptions,

it

In addition,

as an ellipsoid.

it

is

is

assumed that no serious loss

neglected as a dynamic force.

With

apparent that the satellite may be regarded An additional simplification in the

as movlg in a Keplerian orbit. reduction of the tracking data is system exceeds 100 megacycles;

warranted if

for it

the frequency of the

then becomes feasible to neg-

lect both the atmospheric and ionospheric refraction of the transmitted

signal.

In formulating the problem matieatically, the instrumentation as an interferoeter.

it

is helpful to regard

In this sense, the total

number of Doppler cycles observed within any time interval will provide a measure of the change in if

the transmitter is

slant range from the receiver to the satellite

air-borne,

or in the sum of the slant ranges from

both the transm-itte.' and receiver to the satellite if nates on the ground and is satellite.

the signal origi-

either reflected or retransmitted by the

Assuming the latter for the discussion which follows, let

be defined as the change in the sum of the two slant ranges.

g

(t)

It

follows fra

Figure 7 that (t 12

where T is

)

S+Rs) -

-

(+

the position of the transmitting sit-, R,

Ith recriver,

S.

I(1) the location om -=ae

u position of the satellite at time t,, and S

15

the

PROBLEM GEOMETRY FIGMJE 7

16

position a!- +ine t 2 .

gj (t

) is

12

th- cbnnge in the sum of the raant

ranges from the satellite to the transntter and to the jth receiver in the time interval from t this time interval is

to t

It

is

worth noting that, if

equal to one second and

the transmitted signal,

[g j

(t

)

12

" +

X is

frequency for the jtth receiver at the time,

.

is

the wavelength of

equivalent to the Doppler (t

+ 0.5 sec.).

I

The mathematical development )f the computing procedure has been presented in reference 3 and will not be repeated here. Rather, we will confine our remarks to a sumry of the more important phases of the method.

The solution consists of improving a set of position and

velocity components which have been approximated for a specific time.

The latter will be defined as t o and in general, will be within the time interval over which observations have been recorded.

The computing procedure is

outlined in Figure 8.

Initial approximtions for

position and veloci.ty uniquely define a Keplerian orbit which may be described in tems of the following orbital a

semi-m&jor axis,

e

a. eccentricity,

a

a

mean anomaly at epoch,

i

j

inclination, right ascension of the ascending node,

nc

-ters:

a

argument of perigee.

After these parameters have been determined, satellite, and then g3

time.

the position of t tc

(t), may readily be computed as a function of

Comparing the computed values of gj (t)

with the observed values

of the same quantity and assuming more than six observations, a set of differential corrections for the initial

approximatin.- cr

position

and velocity may then be obtained from a standard least-sqares treatment of the resulting over-determined system of equations. are applied to the initial until convergence is

The correctinis

approximations and thc computation is

achieved.

17

ite&ated

U) 0o

0

w

I

A4

0

CL

w 0

z

a

-

2

I--M

0 b I.&1

a

00

0

0* It

0

S

0

This computing procedure essextis.aly determiies oly those segments of the orbit confined within the intervals of observation.

By

constraining the satellite to Keplerian motion, the parametersaa, e, a, i,

n, and w are likewise determined in the course of the computation;

and these serve to provide an estimate of mot~.on over the entire orbit.

On the other hand, it has been found Iqpractical to fit

an entire ellipse

to the observations by solving fc- the oibital parameters directly.

19

n'TTTAL ORBITAL APPOX &TI8NS .

Convergence of the computation resta primarily upon the adequacy of the initial

approximations for position asd velocity.

It

has been

establibhed that, for a system consisting cf a single reeeiver and an earth-bound transmitter at opposite ends of a 400 mile base line, convergence is

assured when the error in each coordinate of the initial

estimate is

not in

excess of 50 to 75 miles and the velocity components

are correct to within 1/2 to 1 mile per second.

When the signal source

is

impossible with obser-

carried by the satellite a unique solution is

vations from a single receiver.

However,

are available from two or more receivers, a.

if

single pes

measurements

with either a ground-based or

air-borne transmitter, the system geometry is greatly strengthened.

Convergence anii then be expected when the initial

apwimtions are

within 150 to 200 miles of the correct value in each coordinate and 1 to 2 miles per second in each velocity component.

larger errors my

occasionally be tolerated, but the figures presented are intended to specify limits within which convergence may be reasonably assured.

Therefore,

it

has been necessary to develop a supporting ccuputa-

tion to provide relatively accurate initial

approximtims to position

and velocity for the primary computation.

Several successful methods

have been developed for this phase of the problem;

but discussion will

be confined to a few applications of a differential equstion, derived, in reference 3,

to approximately relate the motion of the satellite to

the tracking observations.

If

the transmitter is

earth-bound,

this

equation In of the form

A -,2 S

-.

(2)

+

where the slant ranges from the transmitter and the receiver to the satellite are respectively pT and pi. of

t

It

was ass-li

Wj is

he function dpfinel by equation (1). that:

20

the secon

time derivative

In derivinx equation (2),

1)

in angular measurement, the ek,tAlite is within ten degrees of the instrumentation site,

2)

the Earth is not rotating,

5)

the satellite moves in a circular orbit.

With these assumptions,

A may be shown to be appraximately equal to

v 4/(GR) and hence, constant for a circular orbit since R is the Earth's radius, v is the velocity c

the satellite, and G is the mean

gravitationajl constant.

The first application to be considered will be for a system in which the transmitter is carried by the satellite.

For the jth receiver

in such a system, equation (2) reduces simply to

Aj

2

(3)

.j If measurements of the rate of change of the Doppler frequency, fj, are made for two different times, to and t1 , and Doppler frequencies, f, are observed at regular intervals between to and t , we note that (t

~

f(t) 0'

Pj (t) (t 1 o

-

xr

Pj (t1 )

-

PJ (to ) +

*

(t0 ),

t, 1t)

f t

where X is

the wavelength of the siguaL

only unImowns.

f

(t)dt,

0 and, p3

(t1 ) and p i (t,)

are thn

Combining equations (4) with equation (3) yiclds

t5

(t,2 Pj (tc)

-

~ (t 1 2 + Ff' L

L

(to 21

t

(t1 )j

r0

j(t)dt!

which vith the last relation of equations (1) as a fimction of time.

determines elant rang to

()

used with equatio

These results mu¢- *.

establish a value for A from Vhich an excellent approxaimtion of the velocity of the satellite

mey be obtained.

No additional InformLtion can

be extracted when observations are limited to those from a mingle receiver.

However,

if

measurements f

three or more receivers overlap in time,

a set of approximations to the position and velocity components determined by a straightforward trinculation procedure.

Wy be

When data from

only two receivers are available, an estimate of position and velocity may still be obtained for a time which lies within the interval of observation of both receivers,

if

the results of the corxtation described by

equat.on (5) are coabined vith the ass mption of circular notion.

For an

epoch time, selected so that the satellite Is near the zenith of the instrumentnUon site, we mW

of velocity Is sll

safely assume that the vertical component

and can well be approximated by zero.

Using the

reaults of the computing procedure described above, slant ranges for the epoch time may be computed for each receiver; and in the process, 4n estimate for the velocity of the satellite will be obtained.

Coabining

these three results with the Doppler frequency meanurments fr

the two

receivers for epoch time, we my readily determine the remain'n

velocity

components and anl three position coo-dLiates,

in this development, no

account has been taken of the difference In frequency between the transitter

In the satellite and the reference oscillator on the ground.

If

both are stable, a constant frequency error, or bias, wll be introduced. n general, this error is so large that it the above procedure.

Moat methoda,

must be corrected before applying

for determining the bias, assum

ayi*try about the inf'.ectlon point and use this characteristic of the "S* curve to determine the inflection time as accurately as possible. the latter Is also the time of closest approach of the satellt receiver, the Doppler frequency should be zero.

9Therefore, the basa is

simply the observed frequency at the inflection time.

22

Sance

e to the

The second application considers a _7y3tem in which the transmitter is

earth-bound so that the signal travels fron the EaFth'

surface to the satellite and back to one ur more receivers on the

Earth's surface.

For thLs problem, equation (2) applies.

Let us

define a right-hand rectangular coordinate system as shown in Figure 9 with the origin at the transmitter and the Z-axis positive in the direction of the vertical.

The y-axis is

formed by the intersection

of the tangent plane at the transmit ter with the plane determined by the transmitter, the Ith receiver, and the Earth's center. ceiver will then be at the known pcint (0, point (x,

y,

Yji, zj).

If

The re-

the variable

z) indicates the position of the satellite, the slant

ranges from the transmitter and the Ith receiver are respectively given by +y +

T

+ z

,

(6) P

-

l

2

+

)2

(x

Y-yj

+

2) J 2

z-

from which it follows that * +. Yy + z

AT

PT(7)

xi + (y - y ) y + (z pi

j

-.zj)

.

In

the three-beam mode of operation, the satellite will be approximately

in

the yz-plane at to,

which is defined as the tine halfway between the

initiation and termination of tracking in the center beam. satellite's and (*

Let the

position and velocity at this time be defined as (xo1 , Yo, o'

o),

respectively.

Obviously, x

end as before, 1'omay also be met equal to zero. to

23

zoj

may be approximted by zero Equations (7) then reduce

zV

CTE *EOITP ot coxiFIUR

FO0DE0RMNIN N APR XMA

sNoIA *I5,Y~j

째To

=

2Y + Zo0

(8) (YoYj) ko

=

(-ZJ)7j

P~~o /(Yo-yj)f+

Let fjo and fJo be the Doppler frequency and rate of change of frequency for the Jth receiver at t

It

.

follows that

fJOTo fo (6

j)

+

(9)

From equation (2), we conclude

A

2

A

I POoTOoJ

jo

2 (10o)

jo

+

Expressing equations (9) and (10) in terms of the position coordinates

and velocity compcnents of the satellite at time, to, yields

~JOXF~

v

-

+

A-ky2

'y

A

O021 2

JYo

+

XVy 0 + z o

-o.L ) 1 ,) )

(

2

=

jo"

(yo - YJ)(

(%.,

A[

+

(Y'

(3o" 2

+

2

Y() ko

j

2

(째z,

Z) V~ o . )X X" Y-j7 " + (2.o'b

Let us assume a specific orbital inclination.

(12)

With our previous

assumption of circular motion, jo may readily be computed as a fVm-tion of Yo and z o

.

Than

equations (11)

25

and (12)

will likewise provi~e fjo

and fo

as functions of position in the ,z-Plane.

inclination,

Thus, for a given

families of curves may be computed and plotted in the Figure 10 presents such a plot, for an

yz-plane for both fjo and f

incliration of 80 , with the transmitter and receiver separated by 434 miles and with both located 350 off the equator. and simplify the construction of such charts, which is

To attain symmetry

z3 was assumed to be zero,

a reasonable appruximation for this approach to the problem.

If similar charts are prepared for a number of inclinations, satisfactory initial approximatlons may be rather qLckly and easily obtained by the following operations: 1)

Assume an inclination. selecting a chart.

This, of course, is equivalent to

Accuracy is not casential at this stage

slAce the estimate may be in error by 150 without preventing 2)

convergence.

Enter the chart with the observed values of f

and fi to

determine an appropriate position within the yz-plano.

3) Approximate the velocity components.

These should be consistentt

with the assumption of circular motion, the height determined 4)

in step 2), and the assumed inclination.

Determine the position and velocity components in the coordi-

nate system for the primary solution by an appropriate coordinate transformtion.

In addition to the graphical method, a digital solution has been devised for equations (11) and (12). As in the previous development, we have two measurements available and desire to determine three unknowns.

In this approach, one unmown is determined by establishing an upper bound and assuming a value which is a fixed distance fro this bound. The distance has been selected to place the variable between its ,;per and lower bounds in a position which is favorable for convergence of the primary computation. In this method, we chose to start by approxiiating z0 . It may be observed in Figure 10 that, for larger values of it3 o the maximum value or zc occurs above either the tr'mumitter or receiver whiie,

26

-

SIX

30 CPSAS

40 CP/

.j

4

x 70

TRANSMITTER Y-AXIS (EAST

OOPLOC FREQUENCY OF FREQUENCY POSITION IN (FOR SD.

IM MILES)

micIVER

AND RATE 0F CHANGE AS A FUNCTION OF THE YZ - PLANE INCLIN..4rION)

FIGURE 10 T7

for smaller values

f f e the maximum value of zo occurs over the mid-

point of the base lÂąne.

The first step in tte computation is to determine

a maximum value for z0 .

To this end, ko is eli-inated from equations (11)

and (12) to yield an expression which varies only in yo and z0 .

A appears

in this expression, but it is also a function of these variables. resulting equation may be solved by numerical methods for z knd then, solved a second time for z

with Yo -al/2Yj.

The

with YO-

The larger of

these results is to be used as a value for (zo)M which is defined to be the maximum possible value of z . Assuming the altitudes of all satellites 0

to be in excess of 75 miles, we may conclude free the general characteristics of the family of curves for f

in Figure 10, that the satellite's altitude

will differ from (zo)M by no more than 100 miles.

Since an error of 50

miles may bc tolerated in the approximtion for tach coordinate,

E Zo)

- 50

i.a suitable value for z0 .

With the altitude thus determIned,

we may solve equations (11) and (12) for Yo and Yo" hence the velocity, will be determined.

With i

00

readily evaluated to complete the initial

In the process A, and

assumed as zero, i

ms& be

approximations which consist of

the position (0, yo, zo) and thc velocity (*o, ko, 0).

It is worth noting

that there is a pair of solutions for y0 and Y " Further, the method does not determine the sign of x

.

If,

in addition, we accept the possibility

of negative altitudes for the mathematical model, we arrive at eight possible set3 of initial

conditions vhich are approximtely syemetrical

with respect to the base line and its vertical bisector,

It

is an interesting

fact that all eight, when used as input for the primary camputation, lead to convergent solutions which exhibit the same type of syimtry as the

ipproximations themselves.

Of course,

it

is

trivial to eliminate the four

false solutions which place the orbit underground.

Aurther, two additional

solutions may be eliminated by noting that the order in which the satellite passes through the three antenna beams determines the sign of i. :a the two remaining possibilities, Jo is observed to have opposite sins. Since the y-axis of the DOFLOC system has been oriented fram west to east, the final ambiguity may be rezolved by assuming an eastward cacm~nent of

28

velocity for the satellite

-

certain.Ly a valid armmpticn to date.

any event, all ambiguity may be roved of one other receiver.

Moreover,

In

rrom the solution by the addition

this would significantly improve the

gecetry of the system and thereby strenithen the solution.

The first

method presented in this section is

a satellite which carries its

own transmitter.

recorded continuously as in Figure la.

intended for use with

These data are generally

The other two methods have been

developed for a system which provides observations of the type displayed in Figure lb where the siga shown in Figure le is

source Is on the Earth's surface.

also for a system with an earth-bound trantter;

and the last tvo methods may be applied to such data if are made in the procedures. observatlow

The plot

minor modifications

Indeed, with any tracking system that provides

of satellite velocity components,

equation (2)

furnishes an

adequate base for establi&hlng an approximate orbit to serve as an initial solution which may be refined by more sophisticated methods.

29

RESULTS OF ORBITAL COGMUTATIO Numerous convergent solutions have been obtatned with actual field

data from a system consisting of a transmitter a. Fort Sill, Oklahoa, and a single receiver at Forrest City, Arkansas. This system complex provides a base line of 434 miles.

In addition, several orbits have been

established from field data for satellites which carried the signal source.

For the latter mode of operation, receivers werve available at both porrest City, Arkansas, and Aberdeen Proving Ground, Maryland, providing a base line of 863 miles. Results will be presented for both types of tracking.

The initial successful reduction for the Fort Sill, Forrest City sstem wa, achieved for Revolution 9937 of Sputnik iII. The D0PrX observations, as well ae the results, axe presented in Figure 11. Measurements were recorded for 28 seconds in the south antenna beam, 7 seconds in the center beam, and 12 seconds in the north beam, with two gaps in the data of 75 seconds.

Thus, observations were recorded for a total of

7 seconds

within a time interval of 3 minutes and 17 seconds. Using the graphical method described in the previous section to obtain initial approximtions, convergence was achieved in three iterations on the first pass through the computing machine.

It will be noted, in the comparison of DOPLOC and

Space Track results, that there is good agreement in a, e, i, and n, parThis is characteristic of the single pass solution when the eccentricity In small and the computational input is. limited to Doppler frequency. Since the orbit is very close to being ticularly for the latter two.

circular, both a and w are difficult for either the DOPLOC System or Space Track to determine accurately. Hoever, as a result of the mall eccentricity, (co + a) is a good approximation of the angular distance alon the orbit from the equator to the position of the satellite at epoch ti-'. .v as such provides a basis of comparison between the two systems. A comparison of this quantity is included in Figure 11. To summarize, vha limited to single pass, single-receiver observations, the DOPLUC 8-sta provides an excellent deterriwnat-In o,f the orientation of the orbital plane, a good determination of tt.e shape of the orbit, and is 'f.ir-to-poor determination of the orientation of the ellipse

30

ithin the orbital plas.

- - -j

II7 -~0

N

.0

0

~

-Z

IL

A

IrI

00 P A00

oa)

in~~~2

w wi

ILI

4pI

II

to ------

w',0

z~s, I c

Although c and w have been accurately Letermlned cm occasion, interim DOPLOC system with its

the

limitations fails to provide consistently Therefore, only a,

good results for these two quantities.

e,

i,

and n

will be considered in presenting the remaining DOPLOC reductions.

The

observations recorded for the Fort Sill, Forrest City coMplex are plotte in Figure 12 for six revolutions cf Discoverer XI including 172, the lad., known revolution of this satellite.

The DOPLOC determined position for

this pass indicated an altitude of 82 miles as the satellite crossed the base line 55 miles west of Forrest City.

A conpsrison of the Space Track

results with the DOPLOC reductions for these observations is presented in Figures 13 through 16.

In addition, DOPLOC reductions havt been

included for three revolutions in which the receivers at

rest

City and

Aberdeen Provg Cround traced the air-borne transmitter in the satellite. presented for six separate

In Figures 17 through 20 a similar comparison is passes of Transit 1B.

In these, all

observations consist of data obtained

b) receivers at Forrest City and Aberdeen Proving Ground while tracking the on-board transmitter. for a reduction based

Finally, in Figure 21, results are tbuated on only seven frequency observations.

These have been extracted from the

complete set of observations previously presented for Sputnik 1I1.

They

were selected to serve as a crude example of the type of reduction required for the proposed DOP1DC scanning-beam system. method is frequency.

quite feasible for use with periodic,

The example shows that the discrete measurements of

Of course, the proposed system would normlly yield several

more observations than were available in the example.

It

is

noteworthy that numerous solutions have been obtained with

field data from a single receiver during a single pass of a satellite.

Further, these measurements have been confined to three short Periods of observation within a two to three minute Interval. spread over greater distances would, accuracy of the results.

of course,

Additional receivers

considerably enhance the

For example, a system with t'ro receivers ane a

2

.Il -,

z•..

hI

-"

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1i

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it_

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110

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3393 Z49

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0

w

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ar

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aa

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w

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04

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z

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I-II-

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-

W

z

w

0

s

>o

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coo

z

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-

ground transmitter would reduce . 'rrOr PrOP~aio in the ccmA~ptIoMS to approximately a tenth of that to be expected for a system ; 4th a single receiver. Removing the restricti on Of single pass determination would further enhance the accuracy of i he results.

COaU"t~ng time$ have been found to be quite reasonable. Convergent solutions have requiredl 20 to I40 ainutee on the Ballistic Research lAboratories. O-RDVAC vhlt:h requires the coding to include floating decimal sub-routines. Mom modern nOlsIes, such as the DNLEM, -nov under constructiOn at the Ballistic Research laboratories, will perfom the Sam computation In 2 to J4minutes.

Bence, It Is realistic to claim that the system potentially has the capabiity of orbit deterninatioU Within, five m:!zutes of the observation time. in conclusion, the method In general =d, therefore, need be confined to neither Doppler fequazneies nor KePlee-Aa orbits, In partlcuALr, if the limitatton of Keplerlmn motion may be retained, onay minor modificatIon is required to use the method 14h all.oer types of satellite observatins

143

ICIOSPREC MMAsR MD.MS Oue of the methods used for studying the ionosphere is

effect on radio waves propagted through it.

from the measurement of its One effect observed is

the increase in propagation phase velocity above

that occurring in free space (i.e. than in free space).

derived

the wavelength is

longer in the medium

At the frequcncies used, 30 to 300 mc.,

the index

of refraction ez' be =pressed as

4o.25 N

(13)

f 2 0

the electron density in electrons per cubic meter an& f 0 is Thus, the index of refraction Is less than the tranmmitt+-d frequency. where N• is

unity in an ionized medium and the aunt is

changed is

by which the index of refraction

inversely proportional to the squa-re of the frequency.

Use is made of the dependece of the refractive index on frequency to determine ionosphere electron content.

Data with which to make the electron

content computation are obtained by measuring the Doppler frequency shift on two harmonically related signal frequencies transmitted front & satellite.

The Doppler frequency, superimosed upon the lower of the two frequencies, is

multiplied by the ratio of the two sina' frequencies employed; and then

the result is

subtracted frm the Doppler observations of the higher fre-

quency signal to yield dispersive Doppler data.

The dispersive Doppler

frequency is proportional to the time rate of change of the total clectr-t content along the propagation path,

0

where IrNedr is

the total electron content along the propasLion path, r;

ThL& a measure of the clatzgv in electron content, over & time interval of interest,

is

obtained by integration of the disp rulve Doppler frequenc.-.

which Is simply a eye?!

count over the specified Interval.

1414

Instrumentation has been develcped for measuring and recording dispersive Doppler frequency.

Satellites carrying radio tzunswitters

whose frequencies are harmonically related serve as signal sources.

at great distances and provide

order to receive signals frcm stellites output data of high quality it receiving systems.

is

In

necessary to us.. extremely sensitive

Narrow bandwidth, phase-locked, tracking filters

used to provide essentially noise-free Doppler frequency data. channels are used to receive the two iarmonically

are

Two

related sign&".

The

multiplication of the Doppler frequency on the lower frequency signal takes place at the output of the tracking filter to-noise ratio. widthi filter

so as not to degrade the signal-

Frequency multiplication prior to the final narrow-band-

would seriously degrade the signal-to-noise ratio of the

Doppler signal.

A special broadband frequency multiplier (ref. 4) has been

developed for multiplying audio frequencies. unique in that it

The technique developed is

achieves multiplication of an Audio frequency Doppler

signal, which varies many octaves, but maintains a sinusoidal output waveThe multiplication factor can be any product of two's %od three's

form. (i.e.

2,

3, 4, 6,

8, 9,

---.

).

This frequency multiplier is basically a

combination of an aperiodic frequency doubler, push-push, bridge configuration tripler circuit.

circuit and a

Anxiliary circuits with functions

of automatic gain control; clipping, differentiation and phase-locked trcking filtering make possible a iinusoidal output waveform.

Figure 22 shows a block diagiam of a receiving system for ionospheric measurements using the broadband frequency multiplier (ref.

5).

Dispersive

Doppler, Faraday rotation and satellite rotation effects on the signal can be separated automatically and directly recorded as nhown in Figure 23.

This is

a portion of a record from an upper atmosphere sounding rocket

flight in which a two frequency transmicter was carried.

Dispersive Doppler data recorded in a form similar to that shown in Figure 23 can be counted to an accuracy of - 0.1 cycle.

The total electron

content can be expresed in terms of dispersive D.ippler cycles as

45

:

u-Il

a.w~~EJjHJj~J

i:J

I

aU *~

a

19-

:.-

I

a a4

0

-~

.3 I'

~

ma

agZ I

t-~

2

*

~a-

--

a-

I

j

0

~Z

Im

g

-

.3

4

1:3 -

I3

[iJI~

3; 3~a,

--

mm

:~*

I

-

e

* -

0 g

S

I4~I 1mb..

g -

z

3

mm

a-

m~

liiia: :3

~:i1

0

2 I -

ma -a S.

L

"LIP

N dr

where

(0

-

(15)

(IF 1) 13.4 x i0

K01 ) is the integrated dispersive Doppler frequency, PI is

the lower transmission frequency, F2 is the higher, K is the ratio F2 /F I .

Consider the Transit satellite (1960 Eta), where F,

- 54 me and F2 = 324 mc.

The incremental change in total electron content for each dispersive Doppler cycle is N-d

324 . .........

3.24 x 108

x

lo-8

" 6.9 x 1013

meter ":(16) square electrons

Therefore. t;,e counting accuracy of t 0.1 cycle represents a measuroing sensitivity to the change in ionosphere electron content of 6.9 x 1012 electrons/square meter.

This sensitivity is high enough to detect s1l

irregularities in the ionosphere.

A plot of dispersive Doppler data and integrated dispersive Doppler frequency is shown in Figure 24. for a pass of the Transit satellite (1960 Eta) on 17 November 1960. Irregular horizontal gradients in the ionosphere are clearly shown by the variations in the dispersive Doppler frequency that are evident during the second half of the satellite pass.

ibis curve normally has a relatively smooth "S"

shape under undisturbed geo.gnetic conditions.

It is of considerable

interest to note that this record was made following the period of an extremely severe geomagnetic disturbance. ditions existed fro

Severely disturbed radio cou.

November 12 through 18.

One of the most active solar

regions observed in recent years was reported by the North Atlantic Radio Warning Service of the National Bureau of Standards. measure of geomagnetic activity) on November 13 w in this solar cycle.

The A-index (a 280, thehi.hest recorded

An A-index of 25 is considered a disturbed condition,

therefore 280 represents an extremely disturbed condition.

An unusua.ly

high magnetic field intensity vas recorded at the .5llistic Research Ibcratorims rmW.#c,,

-cer station on November 12 and 13 Vhich is -'

48

.n 'n

*.

w..

A4

~L..-.4

&SER IV D2P4Z CYLEI1.

4=UiVRiiDO0E /yLS *

49

t

-

Figure

95.

Thus the qualitative agreement between the irregularly shaped

dispersive Doppler curve in Figure 24 and the disturbed ionosphere is

well

established.

The Faraday rotation effect can also be used to determine total ionosphere content.

Techniques have been developed for separating Faraday

rotation effects from satellite rotation effects by the use of opposing circularly polarized antennai. and a sequence of electronic mixers as shown

in Figure 22.

When the satellite spins very slowly, a simpler method of

determining Faraday rotation cycles by counting received signal amplitude nulls can be used. case.

A linearly polarized receiving antenna is

A plot of ionospheric electron content is

used in this

shown in Figure 26,

obtained

by tsing reneived signal amplitude null data from a pass of the Transit

satellite (1960 Eta).

The ccmputation methods of Bowhill (ref. 6) and

Garriott (ref. 7) were used in the earliest studies.

A complete ray

tracing program based on that of Little and Lawrence (ref. 8) is

in preps-

ration to provide more acciuracy and to eliminate several assumptions and restrictions of the early methods.

50

WAL it:mv

~J

ON

±." E

75

.k 13 No. fme

4....... ..

... .. .. ...... ~~~if

WW--

It

*~~

L

LA23P/OUJ3*

U.2.

CONCLSIONS 0

The infomtion on the Ionosphere,

ained by the met. ds Aes-

cribed, makes it possible to correct refr,tion errors and obtain more accurate orbital parameters fru Doppler lata.

An interesting example

'

the ionospheric effect on orbital accuracy was ooserved in the computation of the orbital parameters shown in Figures 17, 18, 19, and 20.

The

computation was first attempted using the complete "S" curve including the relatively constant frequency limbs.

The limbs represent data ob-

tained during the emergence of the satel.ite from the horizon and recession into the horizon.

The orbit obtained was appreciably different from that

published by Space Track.

Another computation was made using only the

center pwrtion of the "S" curve, while disregarding the limbs.

The so-

lution was qxeatly improved and the results agreed very well with Space Track data.

This points out the large refractive effect the ionosphere

iiLroduces &L low elevation angles of transmission. orbital solution can be compute,

Fortunately, an

from Doppler datAL obtained at quite high

elevation angles, thereby minimizing the refractive crror.

A program has been initiated to combine Doppler frequency observations with electron content data in an iterative computing process to Increase the accuracy of tna

orbital determination.

The comutation will be initi-

ated by determining an orbit from the uncorrezted Doppler observations.

The electron content data and this approxlmate orbit will be combined to compute corrections for the original Doppler frequency aesurezits.

Usiag the latter, the process will be iterated until the refractive error has been minimized in the Doppler data and hence, in the conputcd orbital parameters as well.

. B. PATTON, JR.

V. W. RIM"

55

REmFCES 1.

Richard, Victor W. DOPIOC Trackirs .'ilter, BRL Me.-orandum Report 1173, October 1958, Ballistic Resea srch Laboratories, de -n Proving Ground, Maryland.

Dean, William A. Precision Frequeney easurement of Noicy Doppler Signal, BRL Memorandum Report 1.10, June 1960, Ballistic Research laboratories, Aberdeen Proving Ground, Yryland.

Patton, Robison B., Jr. Orbit Determination from Single Pas-3 Doppler Observations, 17E Transactiona on Military Electronics, Vol MIL-4, Numbers 2 & 3, pp 37r-344, April - July, 1960.

4.

Patterson, Kenneth H. A Broadband Frequency Multiplier and Mixer for Dispersive Doppler Measurements, BRL Memorandum Report 1343, March 1961, Ballistic Research laboratories, Aberdeen Provi" Ground, Mar-land

5.

Crulckehank, William J. Instruwntatior Used for Ionosphere Electron D.nsity Meaurements, BRL Technical Note 1317, May 1960, Ballistic Reseerc h laboratories, Aberdeen Proving Ground, Maryland.

6.

Bowhill, S. A. The Paraday Rotation Rate of a Satellite Radio Signal, Journal of Atmospheric and Terrestrial Physics, 13 (1 and 2), 175, 1958.

7.

Garriott, 0. K. The Determination of Ionospheric Electron Content and Distribution from Satellite Observations, Theory and Results, Journal of Geophysical Research 65, 4, April 1960.

8.

Little, C. G. and Lawrence, H. S. The Upe of Polarization Fading of Satellite Signals to Study Electron Content and Irregularltle3 in the Ionosphere, National Bureau of Standards JournlI of Research, v64D, No. 4, July - August 1960.

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