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Produced by the NASA Center for Aerospace Information (CASI)
Final Report 

Mission Defiroition Study for a VLBI Station 

Utilizing the Space Shuttle 

NAS-S-2SS43 

Professor Bernard F. Burke October 12, 1982 
F ina 1 Report 

on the 

Mission Definition Study for a VLBI Station 

Utilizing the Space Shuttle 

under NAS-5-25543 

Submitted by the 

Center for Space Research 

Massachusetts Institute of Technology 

Cambridge, MA 02139 

Professor B. Burk~, Principal Investigator 

R. Doxsey 

GSFC Technical Offi cer: 

Dr. T. Clark, Code 693 

D. Roberts 

L. Ban n i s t e r 

J. Binsack 12 October 1982 
Final Report 

VLBI - UTILIZING THE SPACE SHUTTLE 

NASA Contract NAS-S-2SS43 

October 12, 1982 

The uses of th2 Space Shuttle transportation system for orbiting 

Very-Long-Baseline Interferometry (OVLBI) have been examined, both with 

respect to technical feasibility and its scien ific possibilities . The 

study consisted of a critical look at the adaptability of current technology 

to an orbiting environment, the suitability of current data reduction 

facilities for the new technique, and a review of the new scienc~ that is 

made possible by using the Space Shuttle a s a moving pla tf orm f or a VLBI 

terminal in sp,-ce. The conclusions a re positive in all respects: no technological deficiencies exist that would need r emedy , the data processing 

problem can be handl ed easily by stra.:.ghtforward adaptations of existing 

systems, and there i s a signficant new r~search frontier to e explored , 

wi th the Space Shuttle providing the f irst s tep. 

The VLBI technique utilizes the great frequency stability of moder n 

a tomi c time standards, the power of i nt egr a ted circui try to perform realt ime signal conditi oni ng, and t he a bilit y of ma gneti c t ape r ecor ders to 

pr ovide ess entially e rror- free da t a recording , all of which combine to permit 

the reali zation of radio i nterf erometry a t a rbitra ril y l arge base line s . 

This ha s permi t t ed ground- bas ed radio obs ervers to achieve angular resolution 

a t housand times grea t er t ha n that achieved by OpticJl ast ronomy, and has 

revealed a wide variety of interesting ?henomena. The size of the Earth 

is a fu nda~enta l limitation and the use of an orbiting radio telesco~e allows 

1 
one to pass beyond this barrier. Chapter II gives an overview of the 

method. 

The specific advantages of VLBI in space are given in Chapter III. 

Simulations of OVLBI with various choices of near-Earth orbits are given 

in Chapter IV. 

A single antenna in near-Earth orbit can be used in conjunction with 

ground-based radio telescopes in t '1e simplest realization of OVLBI configurations. This is the mode in which the Space Shu~tle would be used. 

There is, however, a strong impetus to place antennas in much larger orbits, 

and these possibilities are ex~mined in Chapter V. 

A critical examination of OVLBI must examine the effects of orbital 

perturbations, precession, and decay. These problems ar~ examined in Chapter 

VI, in which it is shown that all such perturbations can be co~rected f or. 

The question of the best orbit for OVLBI applicat~ons can be f ormulated, and 

the results are discussed in Chapter VII . It turns out that the common 

shuttle orbit ~ith an orbital inclination of 57 0 is clos e to the optimum 

choi ce . 

The state of technical readiness of present VLBI data acqu isit ion, 

handling, and processing systems is discussed in some detail in Chap ter VIII. 

The conclusions are favorable: the requirements of OVLBI can be met by 

straightforward adaptations of existing systems. 

The state of technical readiness, and t he intense scientific interest 

of the poten t ial subjects of research lead to the conclusion that the next 

phase of work should be the developme~t of actual mission concepts . It is 

recommended, therefore, that mission concept s tudies be carried out for nearEarth orbiting VLBI experiments, including s huttl e-based, space platform-based, 

and fre e-flying missions. The relative costs and benefits of the various concep ts can guide the program development . 

2 
II. A BRIEF DESCRIPTION OF VLBI TECHNIQUES 

The technique of very long baseline radio i nterformetry (VLBI) 

aff ords astronomers their highest-resolution look at cosmic phenomena. 

The basic principle of VLBl is the same as that of onnected-elemen~ 

radio interferometry - the synthesis of a large aperture by the combination 

of the output s of two or more wi dely-separated antennas. 

In connected-element int~rferometry ~he outputs of the antennas are 

brought together in real time, while in VLBI they are recorded on separate 

magnetic tapes at each station and played back together at a later time. 

This: flexibility leads to vastly i nc17eased angular resolution fo:r VLBI. 

Connected-element arrays are limited to tens of kilometers i n size by the 

requirements of real- time data and time-standard t~ansm ssion and by the 

constrC\ ;.nts of geography . For example, the l ongest baseline a t the Very 

Lar ge Array of· the Na tional Radio As tronomy Obs erva t or y (The VLA) is B 

max 

:: 36 km , or :: 61 0, 000 wave l engt hs a t t he typical wave lengt h of >' 6 cm. This 

corresponds to reso l ut i on of 89: / Bmax: 0.35 arc seconds. On the other 

hand, VLBI n~enna s may be s epar at ed by 2S muc h as an ear th diameter , a1-

though consi derably less is mor e typ ical . If we t a ke B : 2R_ : 127.00 km 

max -c: 

as an example a t >' 6 cm t he base line i s :: 22 0 million wavel enghts , corresponding to r esolution of : 1 milliar c ;erund , 350 times better than the VLA. 

In each kind of i n ter fe r ometry the basic measurement for a source in 

a direction § is the correlated outpu t of two antennas separ ated by an 

.... instanr::'lleous baseline B. This is called the visibility V of the source, 

and is a (complex) function of the source brightness distribution on the 

sky, B(0, 6 ), and of the perpendicular component of the baseline, 

3 
a J- = B - (B.§) §. B-LiS usually resolved into components along the 

(projected-on-the-sky) east and north directions; in units of the wave 

length of observation, these components are referred to as u and v. As 

the earth rotates with respect to the source so does the baseline, the 

projected baseline tracing out an elliptical path in (u,v) - space. The 

... parameters of the u-v track depend on the length and orientation of B, on 

the coordinates (0, 6) of the source, and on the time~ included in the 

observation. 

The relationship between the source brightness distribution B(a, 6) 

and the measured visibility V(u,v ) is that of a Fourier transf ormation, 

til til 

V( u,v ) = f dx f dy B(x, y) exp [2~i( ux+v y)] , 

where x is an eastward and y a northward angular displacement fr om the 

s ourc e c enter (0 , 6) . Knowledge of the visi bilty at all (u , v) would enable us 

to reconstruct the source brigh tness from t he i nverse transform, 

til co 

B(x , y) = du fdv V( u ,v) exp [ - 2TIi (ux + vy)] . 

_CIl - 00 

Howeve r , the visibili ty can be measl!r ed only at those (u , v) points which 

corr espond to possible pr oject ed separ ations of the various antennas 

which are us ed . In connec t ed- element a r rays such as the VL.\ , the N a ntennas 

are so arranged that the cove r age of the (u,v) pla ~ which is a f forde d by 

the Ea r th ' s rotation of the N (N - 1) 

2 baselines is very dense out to some 

maximum spacing B lA , and is zero beyo nd . As ~ result, a nearl - complete max 

Four ier i nve r sion of finite resolution is possible . The result ~(x,v ) ( the 

"dirty map" ) is the convolution of the true source distribution B(x , v) with 
, . 

the point response R(x,y) (the "dirty beam") of the observation, which is 

the Fourier inverse of the u-v coverage, i.e., 

M(x,y) s B*R mff dx' dy' B(x', y') R(x-x', y-y') 

R(x,y) = fdu fdvexp (-21Ti (uv + vy)]. 

(Sampled points) 

The beam R is a sharply-peaked function of angular width 68 (the resolution) of 

68 - 'A/B , max 

which, because of the incompleteness of the u-v coverage, possesses a lew, 

non-zero sidelobe level. 

The effect of the beam sidelobes is to mi~ the true source brightness 

at a given point with a fraction of the source brightness of nearby points. 

Because the Fourier inverse of the beam R has zeros (it i3 the u-v coverage), 

a recovery of B using the Fourier convolution theorem is not possible, so an 

approximate deconvolution of the dirty map is performed in the map plane. This 

procedure, c.alled " cleaning" the map, consists of successively subtracting pointsource responses from the highest points in the dirty map, storing the amplitude s 

of these " components" , and then reconstructing the map by convolving these foi nt 

amplitudes with a "clean" beam having the same width as the central peak of the 

dirty beam, but no sidelobes (Ha gbom 1974). Wi th a dirty beam of low sidelobe 

le\'el this non-linear procedur e wo s well, and it is possib Ie to produce a 

"clean map" of a complicated source wh ch has a dynamic range (the ratio of the 

brightest map feat ure to the weakest believable one) close to the limit imposed 

by the noise in the receivers. The dynamiC range typically achieved with the 

VLA, where = 27 so the u-v plane is very well sampled , is several hundred 

5 
to several thousand. However, in VLBI as currently practiced, the antennas 

are few in number (N S 7) and somewhat arbitrarily located on the Ear~h, so 

the (u-v) plane can be sampled only sparsely. The "holes" whid. result in 

the (u-v) coverage have the consequence of a dirty beam with a high sidelobe 

level. Under these circumstances the map cleaning does not work very well, 

and the dynamic range of the clean map is severely l :! mited, often ten or less. 

As we shall demonstrate in the next section, one of the impor.tant advantages 

of a Space-VLBT observation over current Earth-based VLBI is the tremendous 

improvement in the density of the (u-v) coverage which is obtainable for a 

given source, with the result ~f greatly-reduced sidelobe levels, and therefore 

significant~ y improved maps. 

The second difference between connected-element interferometey and VLBI 

concerns the measurement of the phase of a visibility p~inL V(u,v). In 

connected-element interferometry a single clock keeps time for each of the 

antennas, so thp. ph~se visibility points fo r anyone baseline is nirectly 

comparable to that obtatned simultaneously on every other baseline. (Of course, 

the actual measured phase depends on time-dependent instrumental constants 

which must be removed by observa tion of suitable point sour ces, f or which the 

visibility 1s i ndependent of (u,v), and for which the phase is defined to be 

zero). I n contrast, in VLEr each station has a separate clock. These cloc ks 

are s uf " iciently stable that when the out puts of two stdtions a re combined, a 

coherent cross- correlation may be ob tained fo r a time long enough to reach a 

suitable signal-to-noise ratio f or each separate visibility ?o int V(u,v ) . However, 

because of longer-term clock dr ifts, and ot :ler i nstrumental and a tmospheric 

effects, it is not possible to dire ctly comp a r e the phases of visbility poin t s 

obtained on different baselines at the same time , or on a single baseline 

6 
at widely separated times. Thus a direct Fourier inversion approach to map 

mak;.ng is not very useful. The earliest VLBl observations were therefore confined to measurement of the amplitude of the visiuility, and map making was 

reduced to modeling the amplitude of tr.e visibiltiy by the sum of a small 

number of elliptical gaussians. 

Recent ly , more sophisticated VLBI data reduction techniques which do 

make use of the measured phases have been developed. When observations on any 

three baselines which form a closed loop (i.e., from the same three stations) 

are combined in a suitab Le way the net visibility phase around tfie loop is 

es.entially free of instrumental effects. However, the resulting "closu e 

phase" does depend on the structure of the svurce, and provides constraints in 

the source map which are in addition to those due to the amplitude information. 

When a reasonable number of antennas are involved, the closure phases contain 

almost a ll of tt.e phase information that would be obtainable in connected-elemen t 

nterfer~metry (in an array of N antennas, we ge t all but 2/N of the phase 

i nf ormation). VLBI data reduction ba sed on this idea i s ca lled "hybrid mapping", 

and has grear.ly improved the quality of VLBI maps. (Hybrid mapping is closely 

related to the "self-calibration" technique employed in phase-stable i nterfe rometry, i n wh ich an i nitial map of the source is us ed to g~ nerate a model 

visibility, which is in turn compared to the da t a and t hereby used to correct the 

gains of the antennas. This procedure may be itera ted several times, and wor ks 

when there are many more baselines than antennas, so that the problem is we llconstrained . 1 These techniques should be app licabl e to space-VLBI , but their 

use in the case of numerous baselines which r e not sim ltaneo ~ly present has 

not been explored in detail . Of course, the us e of c losure phase wi ll require 

the simultaneous USE' of a minimum of two ground s tations in concert with a 

space VLBI termi~al . 

7 

j 
III . ADVANfAGES OF A SPACE-VLRI STATION 

We have studied the advantages of using one or more spacE" VLBI 

station~ in a variety of orbits to observe celestial objects at various 

cositions in the sky. In his section we discuss the advantages of such a 

VLBI syste~ compared to an Earth-bound system. One of the limitations of 

ground-based VLBI is the lack of large north-south antenna separations. This 

constraint, which arises because of the predominantly east-west distribution 

of habitable places on the earth, results ~ poor north-south resolution for 

celestial objects of law declination. A second limitati on is that most of the 

large astronomical antennas are in the northern hemisphere. making the southern 

half of the sky effectively unobserved by VLBI. The use of a single space 

VLBI station immediately removes the f i rst of these restrictions. and also 

opens up to VLBI all of the southern sky that can be s een from a single (or 

perhaps two) ground-based antenna. The existence of a t least two large entennas 

i n t he southern hemisphere (Parkes 64-m a nd t he JPL Deep Space Networ k 64-m 

i n A'lstralia ) me ans t ha t with spa ce VLBI we can study obj ec t s anywhe r e i n 

t he sky . A thi rd, f undamental, limitation of ground- based VLBI is the small 

number of antenna ~u ns available. This results in a coa r s e and spor adic 

coverage of ~h e Fo ur r ttansfonn (u-v ) space, since the spac i ng be tween 

var i ous u- v tracks is dic t a t ed by the s epa r at ions of t he various antenna s . 

Recent p"roposa l s fo r a dedica t ed VLBI ar :::-ay of appr oxima t ely 10 ant ennas at 

sites chosen to " opti mL:e" the u-v cover age go part of t he way t owards solvi ng 

t his pr oblem, but t he r e sul ts are limited by the s~all number of sta t ions tha t 

i s economica l ly fe asible , a nd e diffic ul t y of seeing a given ~ou ce simul -

t aneously fr om two gr ound io s which a r e f ar apart. 

A single space \~BI station pr ov ides a continuously changing baseline 

to each gr ound statlon being used . Such a space terminal 1n near- ea r th 

orbit c ircles the Ea r h bout 8 times in a 12 hour period , and because the 
ground stations are in different positions relative to the sour~e for 

each orbit, a large number of u-v tracks result {rom each ground station. 

Figure III-l shows the kind of u-v tracks ~enerated by one orbit and by 

several consecutive orbits. As we demonstrate explicitly ~elow, this 

results in u-v coverage which is decidedly 5uperior to that obtainable 

with current ground-based antennas, and comparable in density and superior 

in resolution to that of the proposed dedicated VLBI arrays. 

The fourth fundamental limitation of ground-based VLBI i s that the 

longest baseline obtainable is t he diameter of the Earth. In practice the 

maximum baseline is somewhat smaller than this because of the 2ctual locations 

of Earth-based stations and of the requirement of simultaneous visibility of 

the source at both ends of a given baseline. The latter problem is exacerbated 

by the location of most large ground-based antennas at latitudes between 30 and 

O degrees north. The use of a single space station results 1n substantially 

increased usable baselines even f or near-Ea rth spacecraft orbits. Large r rb ~ts 

accessible to free-flying spacecraft result in dramatic increases i n baseline, 

with concomitant in~reases in resolution. 

Limits on Resolutions Due t o Interstellar Sc~ tter ng 

The ultimate l imit to the resolution of vtBI is se t by in terstella r 

scattering . Measur ements at low frequencies have shown that a point source 

is broacened into a finite disk of size 

8s ~ 150 (A/Ao) milliarcseconds mas) , 

where r. = /(0 - 2), is a poorly- known parame ter, and Ao ~ 368 cm corresponds 

to a freq uenc y of 81 . 5 ~Hz (see Rickett 1977) . Studies of the Crab pulsar 

have shown that ~ 3, so that n < 3 ; a nominal value migh t b n r 2. 

9 
Figure 111-1 . Tracks in (u-v) space produced by one and by several 

~o nsecutive orbits of a SPdce VLB station end a single ground station. 

10 
>------~~~~~~-------

a:

u 

w 

W 

Cl)a: 

0-

~cn 

>--------~-------+-----------------

11 

(

Ow 

~~ 

a: 

() 

W / ) CI) 

--
The results below are quite sensitive to the exact value of ~ (because 

the VLBI frequencies are so f ar from 81.5 MHz), so we present results for 

n = 2 and n = 3. 

The longest useful baseline is the one f or which the angul r reso l ution 

of the ~xperiment, 

e z z 2 x 10 3 A IB mas , r B cm km 

is equal to the scattering disk 8 . Since 

s 

1.1 x 10- 3 A 2 mas 

cm 

8 = s 3.0 x 10-6 A 3 mas 

em 

2 , 

3 , 

the maximum useful basel i- _, expressed i n Earth radii, i s abo ut 

300 - 1 

A ( 2 ) , cm = 

B I RE z 

max 1.1 x 105 - 2 A (n 3 ) cm 

In Table 111- 1 we pre8ent the maximum base line , a nd resulting maximum 

angular reso lution, a s a function of observing wavelength, fo r the ca ses 

= 2 and n = 3, alorg wjth the resolution currently ob t ainable (8CB) f rom 

gr ound - based VLBI, ssuming B = R

E

. For t he ca se 

max = 3 (0 = 3), it is c lear 

that almost llalimited resolu t ion is theoretically possible - inte:rstellar 

scattering falls off with decreasing wavelength so fast (8 ex: )..3) that a t the 

s 

r.igher frequencies used f or VLB1 the scattering is negligible for basel"nes 

which are tens of thousands of Earth radii. However, i f 2(0 = 4) ,the 

sca tteri ng disk remains a constraint fo r baselines of a f ew hundred Earth 

radii, and resolution would be l imi eJ to a f ew hundred times that urrently 

ava i la ble . It might be 3dded that a determination of a can only be made with 

l ong baselines a nd short wavelengths, so space VLB1 wi ll lead to i mportant new 

12 
, 

. 

I == 2 

, = 3 

• 

A( cm) 

e (mas) 

GB 

6 (mas ) 

B (E):) maX 

)

8 (mas) 

B U<- ' max EI 

TABLE III-l 

Maximum Useful Baseline and Resulting Resolution 

50 18 6 2 . 8 2.0 1.35 

17 5.9 2.0 0.92 0.66 0. 45 

2.8 0.36 0 . 040 0.0086 0.0044 0.0020 

6.0 17. 50. llO. 150. 220. 

0.38 - c; 2. 4xlO ~ 

44 . 340. 3100. 1.4xl04 2.8xl04 

13 
• 

• 

information about the interstellar medium. 

In the next section we begin our detailed examination of the propertl ~ 

of space VLBI with a discussion of the spacecraft orbits involved. 

Specification of the Orbit 

The orbit of a satellite about the Earth is specified by six elements, 

three which give the size and shape of the orbit and the satellites' initial 

pos:~icn in the orbit, and three which specify the orbit's orientation in 

space. The semi-major axis a and eccentricity e specify the size and shape 

of an elliptical orbit. The satellite's position in the orbit at an initial 

time t may be specified by the initial mean anomaly o 

M 

o 

2n 

T (t - t ), [modulus ], o p 

where T is the orbital period and t is the time of perigee passage. The 

p 

period of the orbit is related to its semi-major axis by Kepler's third law, 

2 3 4~ a 

G~ 

where G = 6.67 x 10- a cg s is . ewton's constant of gravity and ~E 

is the ~ass of the Earth. Numerically, 

T(minutes) = 1.659 x 10~ ' a 3/2 (km ) 

a( km) = 33 1. 3 T2/3 (min). 

The standard circular orbits available from the Space Shuttl e will have 

altitude h i n the range 27 0 ~ h ~ 400 km, so since the mean r adius of the 

Ea rth i s R = 6731 km, these have periods in the range 89.8 < T < 92.4 min . E 

An Earth- synchr onous orbit of period 24 hour s ha s a = 42200 km, so h = 35800 km . 

A highly eccentric orbit with a=lOO , OOO km n~ e = 0 . 90, correspond i ng to 

Ititudes 3630 km < h < 184 000 km has T = 524 0 minutes = 3

d ISh 20

m

. 

14 

- - ) 
The orientation of the orbit is described by two angles which specify 

the plane in which it lies and by one which specifies a direction in that 

plane. For orbits around the Earth it is convenient to refer the orbital 

plane to the plane containing the Earth's cent~r and equator, and whose 

intersection with t~e celestial sphere is the celestial equator, and to 

choose the fiducial directio~ in that plane to be toward the Vernal Equinox 

y (origin of right ascension, a = Oh). The longitude of ~he ascending node n 

is the angular distance along the celestial equator between y and the ascending 

node of the orbit, NA, the point at which the plane of the orbit intersects the 

equatorial pla~e and at which the projected motion of the satell ite is from 

the southern into the northern hemisphere. The inclination of the orbit i is 

the angle (0 < i < rr ) between the orbital plane and the equatorial plane; orbits 

are called prograde if i < ~ and retrograde if i > ~, corresponding to satellite 

revolution in the same direction ana the direction opposite the Earth's rotation, 

respectively. 

The sixth orbital element is the longitude of perigee w, which specifies 

the location of the perigee of the orbit wi th respect to the Cele~tial Equator. 

It is the angular distance along the projection of the orbit onto the celestial 

sphere from the ascending node to the point of perigee passage, and satisfies 

o < < 2n . When w = 0 or w , the ma jor axis of the orbit lies in the 

cel estial plane of the c elestial equator, whi l e when w = 7T 2" or w = 3n 

-2- the 

minor axis is parallel to this plane. The geometry of orbital orientation 

in space is illustrated in Figure 111-2. 

The velocity v of the satellite in its orbit i s 

v( km / s) - 7 . 91 

( 

2RrE 

15 
I. 

I 

• I 

Figure 1II-2. Geometry of the orbital elements for a spacecraft in 

orbit about the Earth. 

16 
/" 

.,/ 

!!:J 

GEOMETRY OF SATELLITE ORBITAL ORIENTATION 

/" 

/" 

/" 

ORBITAL PLANE 

SATELLITE 

ORBIT 

.,/ 

/" 

/" ////\ 

PLANE OF THE 

CELESTIAL EQUATOR 

rlgur e 111 - 2 

0 0 " ::u 

"'0 c> o -- a t; 

::0 r ' 

to" c ;p 

G') 

r- rrl =i_ 

-<en 

\: 
\ . 

) 

where r is the length of the radius vector between the illstantaneous position 

of t he satellite and the center of the Earth . The maximum velocity occurs 

at perigee, where r is a minimum, 

and is 

while the 

and is 

r

min = a (l-e) , 

v 

max 

minimum velocity occurs 

r = a (He) , max 

(l+e ) v = v min max (l+e) 

kID /s , 

at apogee, 

R 

7.91 E 1/2 (1-e)1/2 lan/so (--a-l He 

Thus the circulac shuttle orbits have v ~ 7. 7 lan/s, a large elliptical orbit 

(a = 100,000 lan, e = 0.9) has v = 8 .7 kID /s, v. ~ 0.5 lan/s, and a 

max m1n 

geostationary orbit has v = 3 . 1 km/s. By way of comparison, the velocity 

of a ground-based station at latitude ~ is that of the rotating ear~h, 

0. 46 cos ( kIn/ s) . 

As we demonstrate below, the fac t that the station velccities approp r iate 

to space VLBI are one order of magnitude larger than those encountered 

in ground-based observations necessitates only minor changes i n the data 

processing, but enormous improvements in the possible u-v coverage. 

18 

.. 

, 

.. 
• 

IV. SIMULATIONS OF SPAC~-VLBI OBSERVATIONS NEAR-EARTH ORBITS 

In order to better understand the properties-- advantages and restrictions--

of VLBI with one or more antennas in orbit we have performed simulations of 

such observations, varying the source position, spacecraft orbital elements, 

and the number and locations of the ground stations involved. In this section 

we present the results for simulations involving a single space-borne antenna 

in near-Earth orbit. 

The dependence of the reSUlting u-v tracks on the relative 30urceorbit orientation is demonstrated by the examples in Figures IV-lathrough IV-Id 

(summarized in Table IV-2). In each case the source is at a z Oh, 0 • +30°; 

referring to Figure 111-2, the source has right ascension corresponding to the 

direction of the Vernal Equinox ('( ) , and is located 30° above the plane of 

the celestial equator. The orbit is chosen to be circular and to have inclination i = 60°; the longitude of the ascending node (n) is varied from 0° to 

270° in steps of 90°. As a result the orbit is "side-on" (Figure IV-Is), 

" face-on" (IV-Ib). "side-on" (IV- Ic), nd "back-on" (IV-1d) to the source 

directions, respecti vely. The u-v tracks ic h result from a 24-hour observation with a single ground antenna demonstrate how these geometrical relationships t ranslate into u-v coverage. It is clear tha t an orbit which is 

exactly "face-on" to a sour ce with coor dinates (a , " ) haS elements 

1 90° - 6 opt 

n = op t 

such an orbit results 1n u-v tracks with the maximal extent both N- S and E-W. 

19 
TABLE IV-l 

Four Orbits Projected on the u-v Plane for a Source at a - Oh, 0 - +300 

Space VLB Terminal* 

Figure Ground Stations i :1 

la Goldstone 

Ib Goldstone 

lc Goldstone 

Id Goldstone 

*1n all cases the space antenna is in a circular orbit at an 

altitude of 370 km. 

~o 
Figure IV-l: Fourier (u-v) coverage and resulting synthesi zed beam from 

simulationg of VLBI observations of a single source (a = Oh, 

f our different orbits (1 z 60°; 

scale assumes that the observing wave lengt h was 18 em (v = 1 .665 GH z ) . 
,-> 

a: 

l: 

> u 

- ,--- . '1 

, -

o 

LJ l / lJt11-l I 

- ,,--,-- . '1 

GOSTN 

SHUTTLE (i "'60~ n '" 0") 

Fi gur IV- l 

0 0 ~ 

-C 

o - o 

::J t-

,0 -0 

C 

J> G) 

rr-t 

.... 0.001 RRCS[CS 

T 
+ 

rt') 

0 

• 

" 

0 

.. 

GIU 

«2 

" 

~ 

~ .-

.. 

<D 

0 

.-

<= 

.. 

0) 

0 

• -

<.:)en 

OI 

en:> 

1-1-

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On the right hand side of each fig ure is a "three-dimensional" plot 

of the dirty beam (point response) corresponding to the u-v coverage shown 

on the left. In these plots the x-axis is East-West, the y-axis is NorthSouth, and the z-axis is the amplitude of the response. 1he angular scale for 

the beam plot will depend on the wavelength used in the particular simulations, 

and can easily be scaled for any desired value. Fo~ Figures IV-l the wavelength used was 18 cm; for the rest of the simulations ~he figure captions 

contain this information. 

A. Simulations for Three Real Sources 

Several simulations of observations of real astronomical sources are 

summar~zed in Table IV-2. In each case a series of several ground antenna 

combinations has been used, followed by a spacecraft simulation a~ploying 

three ground stations. The spacecraft orbit was fixed, and the results 

illustrate the various morphologies of u-v tracks which can result. 

For the source 3C84, a high- declination radio ga laxy, ground-based VLBI 

produces rather good u-v tracks, even with only three US-based stations (Figure 

IV- 2a) . Inclusion of a midwest station (No rth Liberty, Iowa; Figure IV-2b) helps 

to fill in the "holes", and addition of a European statton (Bonn; Figure IV-2c) 

approxima tely doub les t he extent of the u-v cove age. However, use of a single 

space-based anteDna, illustrated in figure IV- 2d, clearly improves both the extent 

and ensi~ y of the u-v coverage. 

26. 
-

TABLE IV-2 

Three Real Sources Observed with Various Combinations 

of Ground Stations and Space Stations 

Source Ground Space VLB Terminal** 

Figure Name a 0 Stations* i II 

2a 3CS4 3

h 16m +410 20' HSTK, GB,OVRO 

2b 3CS4 jh 16m +41 0 20' HSTK,GB,OVRO,NLRO 

2c 3CS4 3

h 16m +41 0 20' HSTK,GB,0VRO,BONN 

2d 3CS4 3

h 16m +410 20 ' HSTK,GB,OVRO 57° 900 

3a 3C273 12h 27m 

+ 20 19' HSTK , GB,OVRO 

3b 3C273 12h 27m 

+ 2° 19' HSTK,GB,OVRO,NLRO 7' 

3c 3C273 12h 27m 

+ 20 19 ' HSTK,GB,OVRO,BONN h :'I

3d 3C273 12 27m 

+ 20 19' HSTK,GB,OVRO 5r 900 

4<1 W49 19h OSm + 90 01 ' HSTK.GB,OVRO 

lo b W49 19h Oam 

+ 9° 01 ' HSTK.GB.OVRO.NLRO 

loc W49 19~ OSm + 90 01 ' HSTK,GB,OVRO,BONN 

d W49 19h OSm + 90 01' HSTK,GB,OVRO 5r 90 0 

* HSTK Ha ystack O~ servator ·, tf ,~. 

GB :-''RAO , Gr een Bank, W. VA. 

OVRO = Owens Vall ey Radio Observa tory , Bi g Pi ne, CA. 

O = North Liberty Radio Observatory , North Liberty , IW. 

BONN MPIfR, Bonn, West Ger.nany 

**In all cases the space antenna is i n a cir cular or bi t a t an al t i tude of 37 0 km . 

27. 
Figure IV- 2. Fourier coverage and resulting synthesized beam from simulated 

VLB observations of the source 3C84. The angular scale assumes that the observing wavelength was 18 cm. 

28. 
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Figures IV-3 are the results of simulations for the low-declination 

quasar 3C273. As summarized in Table IV-2, the antennas involved in IV-3a 

through IV-3d are identical to those used for 3C84 above. Examination 

sho~s the dramatic effect of changing the source declination from 6 - 41° 

to 6 - 2°: ground-ground baselines involving the usual existing stations have 

almost no North-South baseline projection (v). As a result, the u-v coverage 

degenerates into East-West lines, and there is essentially no North-South (y) 

resolution of the source. This is reflected in the extremely broad beams in 

Figures IV-3a through IV-3c. However, just as in the case with the highdeclination source 3C84, use of the same single space antenna produces excellent u-v coverage (Figure IV-3d). [In this case, even though the source 

right ascension (a - 12h 27m = 187°) corresponds to a nt!arly "back-on" view 

of the orbit (0 - 90°), the u-v coverage is good because the difference 

between "face-on" and "back-on" disappears for sources at very low declinations.] 

Finally, in Figures IV-4 we present the results of simulations for troe 

interstellar maser source W49. This also lies at low declination ( 6 = 9°), but 

is about 90° away from 3C273 in r ight ascension. Figures IV-4a through c 

show that, for a ground-based array, poor North-South resolution occurs for 

5 z 9°, although it is not as poor as for 3C273. Figure IV-4d shows once 

again that, even for low-declination sources, inclusion of a single spaceborne antenna results in a dramatic improvement in the density and extent of 

the u-v coverage. . h Comp arison of Figures IV-lc (a = 0 and n = 180° ) and 

JV- 4d shows that in this case (for W49) we have the same sense of "edge-on" 

orbits. This is in fact the case; W49 has a = 19h 08m 2 287° and 0 = 90°, so 

a - 0 ~ 197°, while the example of ;'iB~ e IV-lc had ~ - 0 = -180° - 180°. 

Further comparison of Figures IV-Ie and IV-4d shows how the additional ground 

stations of the latter case result in denseru-v coverage, and thus in a lower 

sidelobe level than in the case of a single ground station. 

33 . 
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Figure IV-3. Fourier coverage and resulting synthesized beam from simulated 

VLB observations of the source 3C273 . The angular scale assumes that the obser i ng wavelength was 18 em. 

34 . 
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Summarizing this section, we conclude that the use of a single space 

antenna, in a high-inclination (i z 57 : ) but otherwise arbitrar~ly oriented 

("\1 doesn't matter much") orbit results in dramatic improvement in the NorthSouth resolution for low-decli~ation sources, and in some increase the EastWest resolution, and in a significant improvement in the density of the u-v 

coverage for all declinations. In all cases the simulations ran for one period 

of 24 hours. Had we chosen a longer period, t~e ground-ground tracks would be 

unchanged. However, because the spacecraft orbital period and a sidereal day 

are surely incommensurate, a second day would produce space-ground u-v tracks 

which differ from those produced the day before. The density of the u-v 

coverage can. in principle, be made as complete as desired in this way. This 

is a significant advantage over ground-based VLBI, and is independent of the 

number of ground antennas involved in a ground-ground synthesis. Orbital precession, considered more fully in Ch. VI, is only a small effect in these examples. 

B. Comparison and Combination with a Dedicated Ground Arrav. 

The idea of an array of about ten telescopes, at carefully chosen locations, 

dedicated to VLBl, ~ Ver y Long Baseline Array, or VLBA), has been discussed by 

groups at NRAO (1981) and at Caltech (Cohen 1980). A proposal to build such a 

VLBA has been made to ths NSF by RAO. In this section we compare the properties 

of such a dedicated array with those of space VLBI. Because these two approaches 

to improving the current abilities of VLBl turn out to be quite complementary, 

we also discuss the combination of a VLBA with a space-borne antenna. 

In Table 1'/-3 we summarize the simulations we have performed using a dedicated VLB array and various space stations. Fictitious sources a t three declinations, corresponding to three equal-area sections of the Northern celestial 

h hemisphere (6 = 64°, 30°, and 6°), were assigned a = 12 -- these hoic~s were 

made to facilitate direct comparison withe the Caltech study. Each source 

was "observed" for 24 hours with a set of ten ground stations 

43. 
TABLE IV-3 

Simulations with a Dedicated VLB Ground Array 

and Various Space VLBI Stations 

<5 h Space VLB Terminal** 

Figure Source(a=12 ) Ground Stations* i n 

Sa +64° Network 11 13 

sb +64° HSTK, GB, OVRO sr 0° 

Sc +6f+o HSTK, GB, OVRO sr 2700 t 

sd +64° HSTK, GB, OVRO 26°t 270° 

Se +64° Network 11 13 57° 0° 

Sf +64° Network tiD sr 2700 t 

6a +30° Network ti D 

6b +30° HSTK, GB, OVRO s7°tt 0° 

6c +30° HSTK. GB, OVRO Srtt 2700 t 

6d +30° Network 11 13 s7°tt 0° 

6e +30° Network tiD S,rtt 270° 

7a + 6° Network tID 

7b + 6° HSTK, GB, OVRO sr 0° 

7c + 6° HSTK, GB, OVRO S7° 270° 

7d + 6° Network tiD S7° 0° 

7e + 6° Network tID sr 2700 t 

*In al l cases the space antenna i s in a circular orbit at an altitude of 370 km. 

**HSTK Haystack Observator y . Westford, MA 

G~ = NRAO, Gr een Bank, W. VA. 

OVRO ~ Owens Valley Radio Observatory, Big Pine, CA. 

Network 113 = VLB Array #13 (Cohen 1980) . 

= optimum value for this parameter . 

= near-optimum value . 

4 . 
the "Network 13" chosen as roughly optimal in the Caltech study - as .well 

as with various combinations of space antenna orbits and ground stations. 

The =esulting u-v coverage and synthesized beams are presented in Figures 

IV- 5, IV-6, and IV-7. In a 11 cases the observing waveleng th was chosen to 

be that of the Caltech study, A 2.8 cm. 

The high declination source (a = l2h, 0 ~ +64°), observed by the dedicated array IINetwork .l3", results in the u-v coverage and beam shown in 

Figure IV-Sa. It is clear that the array functions exactly as designed 

the coverage is very dense, and the North-South and East-West resolutions are 

comparable, as expected for: a high-declination source. In Figure IV-5b we 

contrast this with the results of a three-ground-station, single-space-~tation 

observation, with the orbit chosen to be "edge-on". Comparison with the 

previous Figure shows one advantage of space VLBI - - the maxllnum baseline is 

~bout 50% larger than that obtained with a LBA. In Figure IV-Sc and 

IV-5d we change the spacecraft orbit to have, first, the optima) ,and second, 

the optimal and the optimal i. Comparison of these two Figures with the 

previous one shows that a considerable decrease in sidelobe level is possible 

if the spacecraft orbi t is chosen "properly". However, comparison wi th Figure 

IV-Sa shows that the density of the u-v coverage afforded by an array of ten 

ground antennas cannot be achieved by three ground antennas and a single space 

antenna (at least not in a sfngle 24 hour observation), 

In order to make use of the large baselines available f rom space and the 

density of u-v cl)verage afforded by a dedicated ground array, we have simply 

combined the two. For an "edge-on" orbit, the results are shown in Figure 

IV-5e, and for an optimal n (but not optimal i) orbit they are shown 1n Figure 

IV- Sf . The best fea tures of each approach are realized in their combin tion -- the 

s. 
Figure IV-S. Fourier coverage and synthesized beam from simulated VLB 

h observations of a source at a - 12 , 6 z +64° . The angular scale corresponds 

to an observing wavelength of 2.8 em. (v = 10.665 GHz). 

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maximum baseline is considerably longer than for the dedicated array alone, 

and the u-v coverage is much denser than for a spacecraft plus a f ew existing 

ground stations. An observation made with u-v coverage like that in Figure 

IV-Sf would be comparable in u-v density to that obtained with the VLA, but 

with the resolution appropriate to an Earth-sized antenna! 

For a source at lower declination (6 = 30° ) , the dedicated array does 

nc~ function quite as well (Figure IV-6a). The North-South extent of the 

u-v tracks is lessened (by sin 6 ) and the length of each u-v track is decreased (by the shortened mutual visibility time of a source of lower 6). 

By comparison, space-borne observations with "edge-on" and "f~ce-on" orbits 

have considerably greater North-South u-v extent and longer tracks, as evidenced in Figures IV-6b and c . Once again, with only a few ground stations 

there are large ",",oles" , and the beam patterns have larger side lobe l evels 

than one would desire. However, combination of the dedicated array and a 

space antenna yields excellent u-v coverage, for either an "edge-on" (Figure 

IV-6d) or "face-on" (Figure IV-6e) or bit . In fact, it works f!!Very bit a s 

well a t 6 = 30° a s it does a t = 64° . 

v.Then the source is moved into the third sec t ion of the sky, <5 = 6°, the 

pr oblems with a purely ground-based become severe (Figure IV-7a) . Al though the 

antenna locations were chosen t o yield significant North-South baselines, in 

particular by putting an an tenna i n Alaska, the v- extent of the coverage is 

quite limited. And at this 6 the leng ths of the u- v tracks are highly restricted by the short mutual visibility times fo r two antennas witn a large 

East- West separation . Comparison with Figures IV- Sa and IV- 6a shows that a 

significant degradation in both resoltuion and coverage completeness occurs 

when a VLBA is used at low declination . However, as illustrated in Figures 

IV-7b and IV-7c the resolution obtained from space VLBI is not highly degraded 

59 . 
Figure IV-7. Fourier coverage and synthesized beam from simulated VLB 

h observations of a source at a = 12 • 

to an observing wavelength of 2.8 em. 

60. 

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for low o. This is because the spacecraft orbit is chosen to be highly 

inclined to the celestial equator, yielding a significant North-South 

spacf ~tenna--ground-antenna separations even for a source on the celestial 

equator. (In Section VIII below we discuss the choice of a single "best" 

orbit for pace VLBI. A tentative answer, i = 57°. is the basis for our 

choice of orbit~_ inclin~: ion in this section, and is compatible with the 

orbits obta'nable with the Shuttle.) Combinations of ground array and space 

antenna, -or "edge-on" and "face-on" orbits, result in Figures IV-7d and 7e. 

Es ecially ~or t he latter orbit, the results of such a combination are nearly 

as good as those fer sources at high declinations. 

A quantitative comparison of the beams generated by a VLBA alone. by one 

space Antenna and three round stations, and by the combination of a VL~ 

and a space antenna is made in Figures :V-8 through IV-IJ. The first three 

Figures, IV-8, IV-9, and IV-IO are effectively cross-sections of the beams. 

These are plots of the maximum absolute value of the beam. evaluated in 

ten annuli surrounding the central eak. T i s effectively displays the 

amplitude of t he beam sidelobes as a f unction of angul ar distance from the 

center. 

The root-mean-s quare (i .e., the fl~ tuation about zero) of the beam as a 

function or radius from the central peak is plotted in Figures IV 11 through 

IV-13. For the region outside the central ~Ak we have used the same annuli 

as above to calc~late the fluctuations in the beam or each of the simulations 

presented in this se c tion. In addition, we have plotted the half-width at halfamplitude of each ce tral peak; both major and minor axes are shown i: they 

differ significantly . 

Study of these figure s confirms tha t t he combination of a VLBA and a 

space antenna 's superior i n reso l ution and i n sidelobe level to a VLBA alone, 

66. 
while a space antenna and a few ground stations has superior resolution 

but poorer sidelobe level. The degradation of the performance of a VLBA at 

low source declination may also be seen by comparing the figures . Finally, 

they also demonstrate that an optimum VLBA-plus-space antenna combination does 

not suffer such a fall-off at low source declination. 

67. 
Fi~re IV-B. Cross-section in annul i of the absolute maxima of the 

beams i n Figures IV-Sa ("Network 1113") , IV-Sc ("Shuttle + 3 stations "), 

and IV-Sf ("Shuttle + Network 1113") . 

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OF POOR QUALITY 

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Fi ure IV-8 

69 . 

3.0 
Figure IV-9. Cross-section in annuli of the absolute maxima of the 

beams in Figure IV-6a ("Network/113"), IV-6c ("Shuttle + 3 stations"), 

and IV-6e ("Shuttle + Network 113") . 

70 . 
Figure IV-IO. Cross-section in annuli of the absolute maxima of the 

beams in Figure IV-7a ("Network 11 13"), IV-7c ("Shuttle + 3 stations" ) , 

and IV-7e ( "Shuttle + Network 11 13") . 

72 . 
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NETWORK #13 

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Figur e IV- LO 
Figure IV-ll. Root-mean-square beam as a function of the angular 

distance from the central peak, for the parts of the beam outside the 

peak for the cases presented in Figures IV-5 (6 - 64°). The horizontal 

bars with tick marks show the half-width at half-amplitude of the central 

peaks; where there are two ticks, they correspond to differing major and 

minor axes. The labels A - F correspond to the Figures IV- Sa through IV-Sf. 

74 . 
0.1 

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B 

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A 

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3.0 
Figure IV-12. Root-mean-square beam as a function of the angular distance 

from the central peak, for the parts of the beam outside the peak in the 

cases presented in Figures IV-6 (6 - 30°). The horizontal bars with tick 

marks show the half-width at half-amplitude of the central cores; where 

there are two ticks, they correspond to differing major and minor 3xes. 

The labels A - E correspond to the Figures IV-6a through IV-6e . 

76 . 

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3.0 
Figure IV-l3. Root-mean-square beam as a function of the angular 

distance from the central peak. for the parts of the beam outs de the 

peak in the cases presented in Figures IV-7 (6 - 6°), The horizontal 

bars with tick marks show the half-width a t half-amplitude of the central 

cores; where there are two ticks, they correspond to differing major and 

~i nor axes. The labels A - E correspond to the Figure IV-7a through IV-7e. 

78 . 
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•• • '. d 

.' . . . '., . . 
V. SIMULATIONS FOR LARGE ORBITS 

In the previous section we demonstrated the ability of a single space 

antenna in a near-eartt orbit to increase the density and extent of the 

u-v coverage of a VLBI observation. The resolution increase of about 50% 

over purely ground-based observations was due to ~e increased mutual source 

visibility achieved by a space-gr0und antenna pair. However, 1n order to 

produce dramatic inc eases in resolution we need to consider spacecraft orbits 

which extend f a r from the Earth. We saw in §III that interstellar scattering 

will limit the longest uSEful baseline, but that even a pessimistic estimate 

suggests th~t base~ines of several hundred Earth radii will be useful. In 

this section we examine briefly the properties of a deep-space VlB system involving two or three satellites in highl -elliptic l orbits. More than one 

satellite is necessary because when the orbit ze is much greater than an 

Earth diameter, all Earth-spacecraft baselines are approximately the same, 

~oughl y the shape of the orbit itself. Thus filling the u-v plane depends on 

spacecraft-spacecraft baselines. In order to be i ble to do so for sources in 

any directIon, the orbits need to be properly oriented. In what follows we 

study the use of two or three orbits of ~ mi-major axes a ~ 100,000 km [so the 

longe st Earth-spacecraft baseline is B ~ 2a ~ 30 R

E], oriente so ~! " the 

max 

spacecra ft -spacecraft baselines have reasonable projections perpenciicular to 

any source d1rection . Iu all cases the angular scale i s appropriate co a 

wavelength of A 18 em. 

A. Two Large Elliptical Orbits 

Two spacecraft orien ed so lha t their major axes are both in the celestial 

equator and are mutuaJly perpendicular are used in the first simula~ions . One 

has its minvr 3A1S in tt~ plane of the celestial equator, the other has its 

~i nur axIs ?erpendicular to t is plane . Th~ two semi-major axe3 are 

80. 
chosen so that the time 6T required f or 10 orbi ts of the satellite in the 

smaller orbi t corr esponds to that for 9 orbi ts of the other one. Three ground 

stations spaced roughly around the Earth were used, so that there were a lmost 

always two spacecraft-ground baselines. Sources at seven locati ons around 

the celestic:l sphere were "observed" for this time 6T. These simulations are 

summarized in Table V-1, and the results presented in Figures V-1a through V-lg. 

With only two spacecraft, it i s not p0ssible to produce good (u-v) coverage 

for sources at all (a, o). This is because orbits of high eccentricity, required 

f or practical launches from the near-Earth environment, are " l ong" in only one 

di r ection. Thus two orbits can span a given plane, but sources 1n that plane 

wil l have u or v limited to the minor axis of one of the orbits. This is amply 

illustrated in Figures V-Ia, ld, and 1f, wher e sources i n the celestial equator 

are "observed" with two orbits whose ma j or axes l i e i n that plane. At higher 

declinations, however, t he resu l ts a re much better , and since such orbits are 

the energeti cal l y l east expensive , a high- r esol ution VLB based on t his kind of 

orbit mi ght we ll be a good first step away fr om t he Earth. 

B. Simula ticns wi t h Thr ee Spa c ecraf t 

Wi th the use of t hree space craft, i ncluding one whose major axis i s highly 

i c1ined to t he celestial eq uator, i t i s possi ble to get ext ensive (u- v) coverage 

f r aIl sourc e di r ecti ons (eL,) . Simulations ba sed or. the thr ee orbits summarized in Tabl e V- I ar (~ presen t ed in Fi gur es V- 2a t hrough 2g . The source positions ar e the same as thos e used in the two- spac ecraft examples. Comparison 

of the sever al examples shows t ha t a "major-axis proj ect1cn" i s ava i lable in 

bo th u and v f or all S0urce positions , as expec t d . Here we have 6T = 8 

larges t-orbit peri ods ~ 9 middle- or bi t periods = 10 smallest-or bit periods, so 

the (u- v) tracks from space craft to spacecraf t are "c losed" and t hus r epeat 

81. 
. .. 

. 

after T. However, in a real situation, the orbit,l periods would be 

incommensurate, and observation f or extended period~ would e able to "fill" 

the (u-v) plane. It should be noted, however, that ~he time scale is set by 

6T ~ 10(3~3) = 33 days, so such a (u-v) plane filling observations would take 

several months for orbits with a ~ 100,000 km • 

2. 
TABLE V-l 

Simulations with Two or Three Elliptical Orbits 

Orbit a ' km) e i OJ n T(days) --

III 100,000 0.9 0° 0° 0° 3.64 

112 92,000 0.9 90° 0° 90° 3.28 

Figure = V-la lb lc ld l e 1£ .!a 

Q (hours) 0 a 0 3 3 6 6 

o Uiegrees) , 0 45 90 0 45 0 45 

Orbit a (lan) e i T(days) ---

III 100, '10l1 0.9 0° 0° 0° 3.64 

11 2 93 , 00 0. 9 0° 0° 90° 3.28 

1f 3 86,200 0. 9 90 270° 0° 2. 92 

Figure = V- 2a 2b 2c 2d 2e 2f 1& 

cx(hours) 0 0 0 3 3 6 6 

o (degreE:s) 0 45 90 0 45 0 45 

83. -
Figures V-l. Simulations of VLB observa tions with two highly-elliptical 

spacecraft orbits. In all cases the orbits are ~hose describea in Table V-I, 

and the seven source directions are spaced around the celestial equator 

and into the Northern celestial hemisphere in such a way as to ge t a good 

sample of the (u-v ) coverage which is generated . The ~ngular scale assumes 

an observing wavelength of 18 cm (v = 1.665 GHz). 

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spacecraft orbits. In all cases the orbits are those described in Table V- l . 

and the seven source directions are spaced around the celestial equator 

and into the Northern celestial hemisphere in such a way as to get a good 

sample of the (u-v) coverage which is genera ted. The angular scale assumes 

an observing wavelength of 18 cm (v = 1.665 GHz ) . 

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VI. EFFECTS OF ORBITAL PRECESSION. AND DECAY , 

In the ideal case of a satellite subject to only the gravitational 

attraction of a spheric~lly-symmetric planet, the five elements (a,e. .~,i) 

which specify the size, shape, and orientation of its orbit are independent 

of time. However, in the cas.e of a satellite orbiting the Earth at finite 

altitude, several auditional forces complicate the picture and cause all 

of these elements to change in time. Here we will be concerned with only 

two most important forces, (i) the gravitational forces that arise because 

of the non-sphericity of the Earth and because of the presence of the sun 

and the moon, and (ii) the aerodynamic forces due ~o the Earth's upper atmosphere. As we sh~v below, these additional gravitational forces affect n and w, 

resulting in very useful slow c!1anges in orbital orientation. The atmospheric drag, on the other hand, results in orbital decay and the eventual 

reentry of near-Earth satellites, and must the7efore be understood and 

mimimized if a long-lasting ~ months) mission is anticipated. 

The effect of the non-sphericity of the Earth and the attraction of the sun 

and moon is to produce an effectively non-invers-square-law grdvitaional force. 

For near-Earth orbits the effects of the sun and moon are re l atively unimportant,! 

resulting in re gression of the nodes (n < 0) and precession of the apsides 

( . > 0) at these rates: 

Moon Sun 

-l~ OS/y r 43/y r 

w +l~32 /y r -+{)~54/ r 

100 . 
Although these ef:ects would of course be important for the preciee 

spacecraft tracking and location needed for VLBI, they are of little 

importance (or utility) for the overall plan of a near-Earth space VLBI 

mission. The f irst-order deviation from the sphericity of the Earth, 

described by its quadrapole mom~nt (dimensionless) of 

causes regression of the nodes and precession of the aps!des at the rates 

n -

and 

w -

-2i1' 

T J cos! 

2 

2 1 

T -2-

where R is the Earth's radius and T the satellite's peri od. Using Kepler's 

third l aw to eli minate T and i nser t i ng numerical va l ues these become 

w 

R 7/2 = - 10 . 2 (- ) cosi (l 2) 2 a 

revolutions /year, 

= +5.08 ( .!L) '/2 (5cos 2i - 1) (1_e 2) - 2 revolutions/year. 

a 

101. 
For a circular obit w is ir12levant (and undefined), but ~he effect )f 

regression of the nodes is substantial, because it changes the orientation 

of the orbit with respect to ~ne sky. In Figure VI-l we have plotted n 

versus altitude h· a - R for near-Earth circular orbits of various i -

clinations. Note that orbits with i < 90° regr~~d, those with i-90° 

have stationary nodes, and those with i > 90 ' (retrograde orbits) precess. 

The most useful near-Earth orbits for space VLBI have i = 57° '(or i ~ 180° 

- 75° = 123°) in order to optimize the resolution of maps made at all possible 

declinations (see § VII). Orbits available directly from the Space Shuttle 

range in inclination from 28° to 57° (for Kennedy launch). The altitudes 

obtainable depend on payload and on the number of OMS units used (one integral 

unit plus zero to three additional OMS units), and range from ~ 280 to ~ 1000 km. 

The maximum payload faJls roughly linearly frau ~ 22000 kg to zero over this 

altitude range, and depends weakly on inclination, lower inclinations enabling 

larger payloads. Over this entire range of orbits the nodal precession ranges 

from -7.8 revolutions/year (h = 270 km, i = 28°) to - 3.3 revolutions/year 

(h = 1000 km, i = 57°). Ti,us ar.y near-Zarth mission entails an orbit which 

precesses a full 360° in between 47 and III d?ys, and a nominal one does so 

in 81 days. This means t.hat the orbit is oriented more or less face-on to 

every source in the sky several times each year. This is a great advantage 

for space VLBI, f or it means that (1) sources anywhere in the sky can b~ 

well observed with any (high-inclination)near-Earth spacecraft orbit, and 

(2) a given source may be observed with the same optimum uv coverage several 

times p~r year. The second point is of great interest, for some of the most 

fa scinating phenomena observed by VLBI involve rapid time v' riability . 

102 . 
Because of the sensitivity of the nodal preces~ion rate to semi-major 

axis, orbits substantially larger than the ones observed abcve do not offer 

the same advantages. The locus in (i, h) -space of those circular orbits 

which precess (n > 0) or regress (n < 0) times per year, 

n ( h ~2 -cosi - (10.2) 1 +"'R · 

is plotted in Figure VI-2 for several values of n. The curves for n - ± 1 

make a us~ful demarcation between fast and slow precession. At the inclinations best for VLBI, i ~ 57° or 123°, such orbits are at an altitude 

of h ~ 0.63 R = 4030 km. Much larger orbits, such as earth-synchronous 

ones (h ~ 42200 k ), have much slower precession rates n ::: once per 

23 years in the earth-synchronous case). However, there are larger 

orbits which precess at much faster rates due to the effects of lunar 

perturbations. The use of such orbits for space VLBI has been conside red 

by the JPL gr oup, and will not be discussed here. 

The effect of the Earth's upper atmosphere on satellite orbits is to 

decrease all three elements a, e, and i , and the re fore also the period T. 

Because of the extreme sensitivit of these effects co the upper atmospheric 

density, and of the latter's large and unpredictable variations with slow 

conditions, a priori calculation of satellite lifetimes against changes 

i n a, e and i is not likely to be precise. However, the relevant time scale 

103 . 
behaves roughly as follows: 

( ~) A 

h /H 

e 0 

ho 

(1 - 2 ~ ) 

Here Tc is the time for decay to re-entry of a satellite in a circular orbit 

of initial altitude h , (M/A) is the m3ss-to-area ratio of the satellite, o 

and the atmosphere is a9s~ed tc have density scale length H and ground-level 

density Po ' If the orbit is initially elliptical, then the first effect will be 

a circularization of the orbit by . decay of the apogee due to energy loss at 

perigee, a process that always takes longer than the subsequent decay of the 

circular orbit that results. In addition, a satellite initially inc lined at 

li l>o will suffer a decay t~ward i - O. 

The lifetioe of a satellite depends very sensitively on its initial 

altitude. In the range of Space Shuttle orbits the equivalent atmospheric 

scale height is between 35 and 70 km, so that a change in i nitial altitude 

of h re ults in an increase of at least a factor of e~ho 71 km in lifetime. 

o 

Thus the difference between h • 270 km and h - 400 km is at least a factor 

of 6 in orbital lifet~e. The actual lifetimes would depend on the mass-toarea ratio of the specific satellite, but it seems likely that an orb it in 

the 00 km range would be suitable f or a mission of several months duration . 

A lifetime of ears would require either a higher initial orb t or pe riodic 

visits and or bital djustments by the Space Shuttle. 

104 . 
Figure VI-l . Rate of nodal precession for near-Earth orbits for 

various orbital inclinations. 

105 . 
121 I I 

PRECESSION RATES FOR NEAR - EARTH ORBITS 

a: 

<t 

I.&J 1=180°- 30° = 150° ~ 

- a: 4 I.&J i =180° - 57°= 123° - 0-

(f') 2 z 

0 j=90° (f') 0 (f') 

lLJ 

U 

w -2 

a: 

0-

....J-4 j = 57° - ~ - - ~ -6 -

0 

-8 

~11:1: I I I I 

0 200 400 600 800 1000 

ALTITUDE h (km ) 

F1 ur e - 1 
Figure VI-2. Number of complete nodal precessions per year (n) as a f nction of 

orbital altitude (h) and inclination (i). 

107. 
..... 

o 

co 

:!/ 

90001 , -r----.---.---.--~--~--~--~--~~ 

8000 

7000 

5000 

h(km) 

4000I-n=-2 

3000 

2000 

1000 

ORBITAL PRECESSIONS 

PER YEAR 

o 10 30 50 70 90 110 

i (DEGREES) 

'i gure v,-

n=0.5 

0=2 
VII. CHOICE OF A "BEST" ORBITAL INCLINATION 

In this section we address the question, "Is there a single optimum 

orbit for a near-Earth orbiting VLBI station?" From the results of §IV 

we know that for a given source position, an orbit that is face-on to a 

given source produces the overall superior synthesized beam. In addition, 

from §VI we know that near-Ear th orbits precesses with respect to the stars 

a few full revolutions per year. This means that the optimum n for a given 

source wi1! occur several times each year, and therefore the specification 

9f the initial ~a1ue nO is unimportant f or missions lasting a few months or 

more. Thus the orbital inclination i is the only degree of freedom, and the 

question becomes "Is there a single inclination i that produces the "best" opt 

overall results f or sources at all possible locations ?" As we show below, for 

a very resonable definiti on of "best overall results" the answer is in the 

affirmative , and the value of i that we obtain is surpri singly convenient. opt 

For a source at given coordina tes (a ,e) , a f ace-on ne a r-Ea r th cir cular 

orbi t has the el ements 

= a + 

iT 

2 

The angle between t he norma l to this best orbit f or this source and the normal 

to an or bit at inclination i, a t the time when n = n I is 

109 . 
'IT 

2' 

The result of i + ~ is that the North-South extent of the synthesized beam 

made with such an orbit will increase as the v-orbital projection decreases: 

06 a llcos n = l/sin(i + 6). ns 

On the other hand, as long as n = n

a , the east-west beam width will not 

depend on i. Thus the solid angle of the beam will vary with (i,O) as 

::n ex: Oe~S a l/sin(1 + 0) . 

If we now define the "quality" of the beam to be inversely proportional to 

its solid angle, 

Ql - l/c; n ex: sin(i + 0) , 

so that Ql is proportional to the number of beam areas in a synthesized map. 

we may find an overall optimum inclinacion i by maximizing the opt 

integrated quality of all the possible maps we could make with respect to 

i . Since the number of sources of declination 0 is proportional to the 

110 . 
area of sky at that declination, so that 

dn dO ex cos o. 

the "integral quality"of maps made at all possible directions on the sky 

is proportional to 

12 

do dn 

do == J 

rr/2 

dO cos <5 sin (i + 0 ) • 

o 

The condition that there be a single optimum orbital inclination i s that 

= II "\ . o ~ 

Using the identity 

= 0 t < 0 . 

sin(a + b) = sin a cos b + cos a sin b, 

we easily find that 

II (i) 

TI /2 n /2 

= sin i f 0 cos 2 0 do + cos i f 0 

TI == -

4 

I sin i + "2 

111. 

cos t. 

sin <5 cos C do 
Thus the condition of maximum becomes 

( ::' ) i opt 

or 

i = oPtl 

Since 

iT ,. 4 cos 

tan-l (2) 

11' - -4-

i opt 

:: 57~52 

sin i -

1 

2 

1 

2" 

sin i opt - 0, 

cos i, 

it is clear that a maximum is achieved, since 

< 0 , o < i < 2 

A most remarkable fact about this result is that such an orbit is just 

the orbit of largest inclination obtainable :rom a Florida launch of the 

Space Shuttle. In addition, and more importantly, the curve I (i) is quite 

1 

broad about its peak. This is illustrated in Figure VII-I, where we plot I (i). 

1 

In fact, any inclination in the range 

32?O~ i < 83~5 

112. 
has 11 within 10% of its maximum value. Thus the "integrated quality" of 

maps made at all locations on the sky is not very sensitive to orbital 

inclination, as long as it is above about 30°. 

In Figure VII-1 we also plot the ratio of N - S beamwidth f or a source 

of declination 0 using an orbit of i to the possible N - S beamwidth opt l 

(that obtained at i = 1

0 • /2 - 0). 

Gns (i, ' o) 

G

ns (io, 6) - 1 

sin (i + 0) 

as a function of 0 for several i. In the case i = i ,:::: 57~52 the N - S optl 

extent of the beam is never more than (1 + Z2) 112 ~ 1. 86 times its minimum. 

2 ..,.2 1/2 

At the important region near 6 :::: 0° the ratio is only -- (1 + --4--) :::: 1 .19. 

Were we to choose i = 45°, the maximum increase in N - S beamwidth is 

l/cos 45° ~ rz :::: 1.41 (occuring at 6 = 0 and 6 = 90°), and the integrated 

quality I(i = 45°) is only .4 percent less than the maximum I(i = i

opt). 

We see that excellent results are obtainable from a wide variety of orbital 

inclinations. 

An alternate definition of quality wt,ich weights smaller beams more 

heavily, 

-2 2 Q2 ~ (solid angle) ~ sin (i 6), 

equivalent to the square of the number of picture el ements, 

113. 
leads to 

TT/2 

o 

changing variables to 

x :: i + 0 

and using the identity 

cos (a-b) = cos a cos b + sin a sin b 

we find directly that 

I__ 

ai opt = J 

i+ TT /2 

dx {Cosi[Sinx cos 2 x] + sini[sin 2 x cosx ] } . 

i 

The condition aI/ oi :: 0 can then be reduced to opt 

3 I' = 3 3 4 2 2 cos i sin i + sin i cos i + cos i - sin4i - 0, 
which is equivalent to 

tan

4 

(i) - tan 3 (i) - tan (1) - 1 = 0 . 

The only physical solution of this quartic equation is 

i opt 2 

- cos 

-1 L-Ls - S \112 

,'-10' ) ::! 58~28, 

which corresponds to a maximum of 12 , In this case the peak of I 2 (i) is not 

quite as wide, but I remains within 10% of its peak value for 

~S < i ~ 9~0 (see Figure VII-i). The ratio of beamwidth at inclination 

i to that at the best inclination for a given source is very similar to 

opt 2 

that for the other choi ce of Q, and has a worst value (at 

( 10 

5 - 5 

1/ 2 

) 1 . 90 . and an equatori al (6 = 0° ) value of 

We conclude that any orbi t with i nclinati on s a tisfying 

40 ° < i S 6.0 ° , 

with a sl igh t pre fe rence for 

10 ) 

s + Iswi ll yield excellent overall r es ults, and wi ll not severely compromise any 

region of the sky . 

llS . 

:: 1.18 . 
Figure VII-l. Integrated "quality" of synthesized maps of sources 

distributed over the sky, as a function of the orbital inclination' 

of the spacecraft VLB terminal. Also shown is the ratio of North-South, 

to East-West beamwidth for synthesis performed for a source of declination 

o u~ing various satellite inclinations i. 

116 . 
ORIGINAL PAGE ,. 

OF POOR QUALITY 

_ 1.0--..---.......---........--------..----.--------- .... 

)-

t-

::J O.S 

<t 

::> 

:0 0.6 

0

0 10 20 30 40 50 60 70 SO 90 

i (DEGREES) 

en 

I 

Z 

~ 

~ 1.6 

CD 

........ 

~ 

<l: 

UJ 

CD 1.2 

(f) 

. 

. . 

.... i = 30° . . . . . . 

. . . . 

. ,i = 450 •••••• 

, -.. , ... , ... " " ". ... '-~ ". '0 

i = ioPt 2 = 5S~28 

i = iopt 1 = 57~52 

............... . ... ...... . ..... 

1.°0 10 20 30 40 50 60 70 80 90 

8 (DEGREES) 

Fi ure II- l 

117 . 
VIII. CONSTRAINTS ON DATA ACOUISITION. HANDl ING AND pROCESSING 

A. Mark III Svstem 

The Mark III system is a digital data recording and processing system for 

VLBI developed jointly by NASA (Goddard Space Flight Center), Haystack 

Observatory, NRAO, and MIT for astrophysical, astrometric, and geophysi cal 

observations. The Mark III system consists of two major components, 1) a set 

of identical data recording stations, one at each observatory i n the network, 

and 2} a special purpose processor, locat ed at Haystack Observatory, f or 

replaying, correlating, and analyzing data. Since an orbital VLBI s tation would 

be used in conjunction with the existing ground systems, it is clear that the 

orbital station should b~ compatible with the standard Mark III system. In 

part icular, the output data stream should be identical i n format to that of 

the Mark III stations, in order that it may be replayed by the processor 

without modificati on or cQdition of extra hardware . 

. B. ~ark III Recording Stations 

The Mark III r ecording stations each contain an IF distributor, 14 video 

converte rs, a dat a forma t ter and a 8 track instrumentation tape recorder. At 

each st ation the IF dist ribut or is suppplied with a stable 5 MHz clock and the 

IF signal from the telescope . The ruilt- in local oscillators in the converters 

have a range of 100-500 MHz in 10 kHz s t eps. E~ch converter produces n upper 

and lower s~~eb3nd output video signal. The video output bandwidth is selectable with 4,2 , 1, .5 . 25 and . 125 MHz being vailab1e . Each video si nal is 

clipped and one- bit sampled at the N quist frequenc I twice the bandwidth . If 

the ou tput voltage at the time of a sample is positiv the bit is a "1" if 

negative, a "0". 

118. 
Each of the 28 di gital data streams is then ent to the data formatter. 

The formatter adds parity bits (1 per byte) and sync blocks. The sync blocks 

contain .ords for identification of ~ blocks, the time of the first sample of 

the block, and an error checking code for the block. These blocks are 

inserted in each data channel, insuring that each channel can be analyzed 

independently. Each channel of data i s recorded on one tack of the 28 track 

tape recorder. Because of the addition of a par~ty bit each byte, the 

recording rate per track is 9/8 the sampling rate. 

C. Mark III Processor 

The data tapes made at each of the rec~rding stati ons are brought to 

Haystack Observatory for processing. The Mark III processor consists of a 

modular special purpose hardware corre13tor and a host computer which control s 

the processing . Tapes from three staticns can be processed simultaneQusly. 

The 28 data streams from each of the 3 t a es are routed i n such a wa ' that 

each of 84 modules rece i ves a as~line-track pa i r of data s treams. In 

general, the data can be processed at the same rate it was 

taken. Each mo dul e cross- correl ates i ts data s t re ams over r elat i vely s hort 

(~2s) i nte rvals , known as parame ter pe ri ods , passing t he results fo r each 

period to a disk file via t he host computer. The host computer ovide~ each 

modul wi t h the app r opr ia t e input pa r ameters fo r pr ocessing each in t erval , namel 

the time delay , delay rate , frin e ra t e and fringe accelera t ion. Upon comple t ion of the correla t ion pr ocessing , t he output recor ds r e sear ched for frin es 

by softwa r e ro utines running on the host compu t e r. The corre~a or processing 

step reduces the ata volume ~y a fa c tor of l05 . The major requirement for 

119. 
succesful fringe search is that the parameters used by th hardwarE correlator 

be sufficiently close to the correct vAlues so that the fringes are not lost 

due to less of coherence or to ~aring in time. 

Figure VIII-l i.s a block diagram of a correlator module. Each of he two 

input data streams is passed through decoders, which remove sync blocks, time 

tags and parity bits, restor~ng thp. data stream to radio s ource data o~y . 

The time tags are used to properly synchronize the two tape drives . Tha 

X-data stream then passes through a pro grammable delay buffer. This buffer, 

using para~cers passed from the host computer, makes the final adjustment 1n 

delay between the two streams to compens ate for the geometrical delay , T

g, 

between the two recording stations. Due to the Earth's rotation, the time 

delay varies during a parameter period. The delay can be incremented or 

decremen t ed by one data sample interval 15 t .mes per parameter period, with 

the update interval specified by the hnst computer . 

The host computer also s' pplies the fri~~~ rate, , and fringe ~cceleration 

to the module. These are then used t o generat~ 3- level approx1mation~ to 

sine and cosine fur_tions with periods equa2 to the frinse period ~hich 8ch 

multiply the Y data s tream. The cosine and slne s tr ams re t~en cross correlated 

with the delayed X s tream at 8 different lags, resulting in compl x cross 

correlations . These ross correlations re accumulated for a paramete r pc iod 

nd the resulting ' IJmB r e outpu t to the hos t computer for stora .. in disk He . 

The rin e pho~~ mode i , and .• mus be s ufficiently accurate that the 

difference between t~e predicted and ctual fringe phase do 5 not chlmg~ b more 

than f r action of a rot tion during a p~ ramet er period . 

120 . 
D. Orbital Stations 

An essential requirement for performing useful VLBI obRervations is 

motion of the receiving antennae relative to one another. In current observations, this requirement is achieved by the rotation of the Earth on its 

axis. In VLBI observations involving one or ~o_p. orbital receiving stations, 

the relative motions will be primarily a result of the orbital motions of the 

spacecraft carrying the stations. In general, this w: 1 result in higher 

relative velocities of the antenna~ and it must be shown what effect these 

will have on the taking and analyzing of VLBI data. Furthermore, in order 

to analyze the data one must know, ~ priori, the relative motion of the antennae 

to certain accuracies. It must therefore also be shown that the relative motion 

of the orbital antennae can be determined to within the required tolerances. 

These tolerances, to some degree, will reflect the capabilities of the processing hardware and the amount of time that is considered reasonable to devote 

to ' analysis of th e data. Iu l section we will discuss these issues with 

regard to the current capability of the Mark III VLBI Processor at Haystack. 

The maximUUl relative velocity of a pair of antennae on the Earth along 

the line of sight to a source depenris on antenna locati on and source direction. 

The :Jorst case maximum relative velocity is 0.92 km/s for ' a pair of antennae 

on opposite sides 0i the Earth, on the equator, observing a source on the 

celestial equat r . Relative velocities ~ 1/3 this vdue are more typical . ' The 

maximum relative velocity that can be obtained with one of the antennae in 

orbit will depepJ on the particular orbit chosen for the spacecr8ft . Spacecraft 

i circular near near Earth orbits (92 min orbital period) have orbital 

velvcities of ~ 7. 7 km/s. Spacecraft ' 0 synchronous orbits have velocities 

of ~ 3 km/s. (If a highly eccentric orbit is used, the majority of it will be 

121. 
at relatively low velocities. When the spacecraft is near the arth the 

velocities may be higher, but this portion of the orbit will be less useful 

for VLBI purposes and could be neglected.) The maximum likely relative 

velocity is therefore 7.7 + .46 ~ 8 km/s. Again, typical values for the 

projected line of sight velocities will be substantially smaller, and will 

average ~ 15 times those obtained during strictly ground based observations. 

Doppler Shift of Bandpass 

The observing frequency for ' each channel is selected by the local oscillator 

and frequency synthesizer at each station. If the stations are in relative 

motion along the line of sight then the passband for one station will be Doppler 

shifted relative to the other. It is necessary to keep the Doppler shift 

significantly smaller than the bandwidth of the channel, or the correlation 

between stations will be lost. The Doppler shift is: 

v V =---c 

v 

o 

where V is the nominal observing frequency and v is the relative line of 

o 

sight velocity. The Mark III system has six allowable bandwidths, B= 4, 2, 

1 , 0 . 5, 0. 25, 0 . 125 lliz, of which 2 MHz is the most commonly used. The band 

shifts by 

v 

C 

1 

c -,,-

o 
so that 

6v (in MHz) iZO.SO.(in em) max 

Thus at IS cm there is clearly no problem as 6v i; only .04 ~lliz, much 

max 

smaller than the available bandwidths. At 1 cm, however, 6v _ .S MHz, and 

max 

no longer negligible compared to the largest available bandwidth. This effect 

could be reduced by varying the local oscillators of the orbital station in 

orderto match a nominal RF passband. This would require somewhat more 

complicated hardware in the orbital video converters and increase the analysis 

complexity slightly. It must be emphasized that the sensitivity loss caused 

by this effect can not be recovered in the post-processing stage, although it 

could be calibrated out. 

F. Correlator Parmeters/Restrictions 

The basic parameters which must be supplied to the co rrelator hardware 

are the geometric delay ( g), delay rate (i g), fringe rate (~) and fringe 

acceleration (~). Vario us hardware limitations i n the correlator will place 

restrictions or requirements on each of these quantities. 

Delay. The nominal delay time must be provided to the correlator so the 

two data streams can be aligned within liB in ordpr to retain coherence . The 

complex correlation is computed for S lags so the nominal delay must be 

accurate within + 4 data samples. One lag corresponds to the data sample 

interval, 1s = l/2B, where B is the bandwidth, so the required delay accuracy 

is ~ 4/2B = + 1.0 s(2MHz/B). The uncertainty in delay, l is entirely due to the 

uncertainties in the parallel baseline component, ln ' so that T = ~1" I c, and the 

123. 
baseline must be known to an accuracy of 

6'''..1\ < 300 (2MHz/B) meters. 

Since the locations of the ground statio:ls ar ~ Nell determined, this requirement 

translates into a requirement on ~~ledge of the absolute spacecraft location . 

This requirement is independent of observing frequency and the nature of the 

spacecraft orbit (high or low ~arth). 

T ~ degree of accuracy should provide no serious d~fficulty. Spacecraft 

in high orbits will suffer relatively small disturbance forces and their orbits 

should be easy to determine. For spacecraft in low earth orbi t, use of STDN 

or TDRSS tracking will be nearly sufficient. The projected errors for STDN 

and pacel~b are at just the required level. Those for TDRSS and Spacel81 

are a factor of 2 too large. At worst this might require an extra pass through 

the data. Once fringes have been found, use of the VLBI system itself with 

bright calibration sources will determine the orbit extremely accurately. 

Delav Rate. As the distance between sta tions changes the delay must be 

adjusted by the correlator. The adjustment can only be made in units of 

single data intervals, 1s ' A total of 15 adjustments to the delay ± 1 sample 

interval each can be made duri ng a correlator parameter period. This is a 

restriction imposed by the correlator hardware. The delay rate is proportional 

to the relative line of sight velocity, 

g 

d 

dt c 

c 

The time , Tu. for the delay to change by one sample, 15' is 

5 = -- = c 

2Bvl\ 

124 . 
Since 15 updates plus the initial value are allowed, the update interval 

lU is 1/16 of a parameter period, lpp, so that the parameter period must 

satisfy 

T < pp 

8e -- = 0.15 ( aVl1 

2MHz ) 

B sec: . 

The parameter period is normally chosen as large as possible, 2s, 

in the processing of standard VLBI data. The primary effec't of using a 

shorter value is to increase the number of parameter periods contained in an 

observation. S ncer there are a fixed number of output words per parameter 

period. this implies that the required disk storage space will increase. The 

worst case is for vmax, which yields a factor of 13 increased storage. This 

should not be a significant problem since the amount of disk space available 

to the nost computer could easily be expanded to more than the -1 M byte 

currently available. 

A second effect of the shorter parameter period is that the host computer 

must sr rice each correlator module more frequently. The rate at which the 

servicing can be done will depend to some extent on the degree of difficulty 

of the computations used to derive the various model parameters. This could 

require restructuring the system to process less than the f ull number of t~acks 

simultan~ou ly. Decause of possible restriction: in the data rate from an 

orbital station, it is not clear ho'w many channels a space system might have, 

so this mayor may not prove to e a conce rn. 

125. 
Fringe Rate. The fringe rate must be supplied by the host computer to 

each eorrelator module. In order for the fringe search routines to find 

fringes in longer segments of data, the total phase rotation between the 

model and actual data must be a small fraction £ of a rotation over a parameter 

period. Since the fringe rate is related to the delay rate by 

VII 8 x IDs 

tfl = TgVO a = A A(cm) 

( v II ) \ 8 km/s Hz, 

the requirement of small phase rotation over a parameter period, 

6<1>= f Tpp 

oq,dt 

6vU ::: 

0 

T pp < E , 

means that the difference be~ween true and model parallel velocities must 

satisfy 

< E 

At 18 em and Tpp = .15, a nominal value of E = 0 .5 requires knowledge of the 

spacecraft velocity to within ~. 6m/s. This requirement can be easily met by 

TDRSS, but misses by about a f actor of 2 fo r STDN. Again, it should not be 

difficult to obtain the orbit to this precision. In particular, for bright 

ca libration sources, it will be easy to .measure fringe rates to better than 

10 mHz, which corresponds to v ~ 0 . 2 cm/s i f A = 18 m and 0 . 01 cm/s 

if A = 1 em. This should result in extremely good orbit determinations. 

The maxi mum fringe rate achievable in the eorrelator is the recording 

bandwid~h divided by K, a software-detectable parameter satisfying K = 16, 8, 

, 2, 1 only, or 

. 

~ I < 2 x l06/K H ~correlator z. 

126. 
Thus for the maximum near,-Earth fringe rate K must be chosen to satisfy 

K < 2.5 A (B/~~z), cm 

so for a 2 '1Hz recording bandwidth K must be chosen according to: 

> 6 . 4cm .. K = 16 

6.4cm > 

> 3.2cm K = 8 

3.2cm > A > 1 . 6cm K >= 4 

1. 6cm > A > 0.8cm K = 2 

0.8cm > A > 0.4cm K =1 

O.4cm > A > 0 .. not possible. 

The fringe-rate resolution of the correlator is 

= sample rate 

224 K 

= 

0.238 

K 

B 

(ZMHz ) millihertz, 

so that the largest possible value of K should be chosen. 

127 . 
Fringe Acceleration. The fringe rate will not be constant; there is a fr i nge 

acceleration of 

<p (Hz/ ) • x • 990 (92 min ) 2/3 _-=1:....-_ 

T I (em) 

In the absence of a fringe acceleration correction a phase error of 

L ~ 2 11.1 

'l '+' l pp - A ( em) 

2 lpp 2 

(0.05 s ) rotations 

will occur by the end of one parameter period. This amount is intolerable 

for most VLBI wavelengths. However, the Mark III system has provision for 

fringe acceleration correction made by the correlator hardware. The fringe 

phase ~ is updated every K data samples by an amount ¢. which is a 24 bit 

word that can be between 2- 1 and 2- 23 rotations and at either sign. This 

phase update itself can be updated every L data bytes (8L data samples) by 

incrementing or decrementing the least signHicant bit by one. Here L is 

only integer between zero and one-eighth of the number of data samples in a 

parameter period. The least significant bit in 6¢ i s 2- 23 rotat i ons, so the 

fringe ac celeration mode i p 

<Pcorrelator = 

or 

= 

+ 2- 23 rotations 

(time for K s amples) (time f or 8L s ampl es) 

2 . 38 x l OS Hz /s 

K L 

128 . 

( fS ) 2 , 

4MH~ 
where fs - data sample rate • 2 x bandwidth of one track. Combining 

this with the required fringe accelerator for near-Earth space VLBI 

results in a requirement on K and L: 

K L • 241. 

fs 

A (em) (4MHz ) 2. 

Once K is picked to satisfy the size of the fringe rate for a given A 

amd fs , L follows from this equation. For example, at A7.8 cm; where 

K • 4 is the best choice, we require that the phase update 6¢ be upated 

every L • 169 bytes, or a total of 443 times in a O.lSs parameter period. 

This is easily accomplished by the current Mark III system. 

Quantization of the phase update 6¢ and of its update leads to an 

accumulated phase error over a parameter period. This may be estimated by 

ct)mparing the quantized "stair-step" approximation to the phase update to 

the true required phase changes. The absolute level of the stair-step 

approximation is certainly accurate to within ~ of the update quantization, 

i.e. I to with n 2- 2 4 rotations. The slope of the stair-step is accurate to 

within one step per parameter period. Thus the i th update (at time t) is 

in error by no more than 

+ ( )i 

pp 

T f 

Since this error is added to ¢ every K data intervals, or ( _ .... P .... p....::;,s_) times 

K 

in a parameter period, the phase error accumulated by the end of the period 

i!; (f s 

1=1 

/ K) pp 

pp 

f K f 2 

2- 23 (_s ___ ) 1/2 ( s pp) 

K 

pp 

= 2- 23 ( f / K) rotations. 

s pp 
Inserting numerical values yields a phase p.rror or 

1 

K rotations. 

Even for the minimum value of K 2 1 this does not present a problem. 

130 .