The results of a quantitative analysis of the returned signal received from a rotating spherical planet are interpreted by the aid of the Cornu spiral. These agree ...
80
PROCEEDINGS O F THE IEEE
Physical Nature of the DopplerFrequency Spectrum from a Rotating Planet and its Effect on Detection*
length of the inner zone strip corresponding to delay depthof X/4. The lengthof the zone strip is (2R cos +)a,,and this is equal to
E)
a
where Q radians/sec is the resolved angular speed of the planet and k 6 is the longitude measured as in Fig. 1 . The implications of this formula are best illustrated by a table of Fresnal zones-take carrier 2450 Mc, x approximatelyequals 5 inches, planet 4000-mile radius, Doppler shifts ( 1 ) rt: 7000 cps and (2) i.100 cps.
10,000
'Os
1
1.25 inches 125 inches 1042 feet '%%es
1
0.40 mile 0.04 mile 0.004 mile O.OOO4 mile O.ooOo2 mile
1
Clearly, no known master oscillator has
the frequencystabilityrequiredtodetect these signals, ;.e., for f ~ k=7000 cps about 1 in 1014, fn = f100 cps about 1 in 10l6. Notefurthertheextremely low difference frequencies thatmustbe considered when squaring a band of such frecjuencies; a correlation study would need days. 2 ) The Cornu spiral of Fig. 2 [l], [ 2 ] is of great assistance in interpreting this phenomenon, and by its use the follow-ing facts of interest were established. a) Referring to Fig. 1 , a narrow strip of longitude 4 and distance x from the axis of rotation as projected onto the visible disk of the planet, hasin the perfect sphere a radiation contribution equal to 70 per cent of the *Received July 1, 1963; revised manuscript re-
ceived July 23, 1963.
a(phase) --
- 0.
The frequency of the phase is now
The results of a quantitative analysis of is thelength,equalinnet effect, to the the returned signal received from a rotating whole strip atlocation x. spherical planet are interpretedby the aidof This result is true to high accuracy for a the Cornu spiral. These agree with the experimental resultswide variation of the laws of reflection in terms of cos c$, so long as the reflection obtained for the moon and Venus, and indicoefficient is uniform. cate that the integration of samples of the b) Referring to Figs. 1 and 2 , for a pass received signal is dominated by the spherical band of Doppler frequencies equivalent to shape of the planet in spite of random surthose arising from the strip between xo+D face irregularities large in terms of waveand x. -D on the observed disk of the planet, length. theinstantaneousvector signal at time f , OUTLINE OF METHOD whenreferred tothephaseplane of fixed frequency Analysis of the reflection from a spherical planet of a CW signal of minute wavelength compared to the radius of the planet w (1 - 2 radians/sec yields facts important in the detection of the signal. 1 ) Whatever the bandwidth selected for is proportional to study in the continuous Doppler-frequency spectrum, it can be represented by a suppressed carrier double sideband modulation of the type where, although the carrier frequency remains constant in the pass band, it suffers a very high rate of change of phase withchange of themodulatingfrequency, given by 2s tan 4 (1)
~~
a) The centrally symmetrical pass band where xo = 0 and
aW
SUarbiARY
0-1 99-100
January
where V ( u ) is the radius vectorof the Cornu spiral; v is the relative radial velocity of the planet, w radians/secthetransmitted fre-
0.57'
4Zi7"
I
0 .i o
-,
sec
quency; x. and D of Fig. 1 are in miles; c = 186,000 miles/sec. I n practice, on-off operation is necessary to make radiometric comparisons of signal and noise, and in any practicalcase of rotating planets QRt is negligible compared with the minimum attainable value of D (which is limited by bandwidth). Further,althoughthestrip 2 0 wide is approximatelytrapezoidal off center,and the analysisbased on a constant return from eachelement of stripwidthequaltothe mean for thebandpass, it canbe shown that the correction termto allow for this taper of intensity is small compared to the aboveintegraland in quadraturewithit, and hence has no practical effect on the results. 3 ) This formula is readily applied to two characteristic cases.
w
(1
-
T),
and the square bracket of ( 2 ) becomes substantially equal to
2v j
Dl
WRC
Theoretically, for extremely narrow band pass with u between 1.0 and 1.4, maximum value of 2 V ( u )occurs at about 1.8. Any increase of bandwidth beyond this causcs the vector to taper asymptotically to .\/2/450, travelingalongtheCornu spiral. Clearly, the variation for a large increase of D and hence of bandwidth is very little. b) Nowconsider thenext off set frequency band of width 2 0 , i e . , of x o = 2 D . Thesquarebracket of ( 2 ) now becomes equivalent to
This oscillatesbetween a minimum of zero and a maximum equal to twice the diameter of one of theinner circles of the Cornu spiral for large u, and hence in general is a t least 15 d b down on case 3 a). I t steadily diminishes as x0 becomes larger.
Example Applying this to Venus [3],near conjunction f ~ 545-/sec, = using channels S ~ / s e c wide, D = 220 miles, (3a) xo = 0, (3b) x o= 440. ;Issume that w=2aX2.5X109 cps, R=4000 miles and c=186,000 miles/sec. Eq. ( 2 ) becomes
+
exp. { -j21.1(0.000261t XO)] [V3.67(d 0.000261t X O ) V3.67(D - 0.000261t - X O ) - 2V3.67(0.0002611 X O ) ] .
+
+
+ +
Case 3a)
Thephase quency w
reference planeis
(1
-
at fre-
F)
radians/sec,
xo=O, D=220 miles, and ( 3 ) becomes
exp (-io) { 2V(3.67 x 220)] --f 42/4s0, i.e., terminates near the asymptotic center of the Cornu spiral. Case 3b) The phasereference plane isa t frequency w
(1
-2
"">
radians/sec.
Correspondence
1964
t
o‘6
0 0
o+
02
0.6
81 the amount of energy reflected back to the observer could vary widely as theplanet rotates. The theory, however, indicates that a regular sampling and integration process could build up a score for reflection from the inner Fresnal zone. In the outer Fresnal zones, smooth surface irregularities from thesphereareunlikely to give rise to appreciable signal owing to the veryhigh rate of change of phase with frequency which prevails, ensuring that even if the “local coils” of the Cornu spiral are considerablymodified, their effect on the instantaneous sum vector over even a narrow bandwidth is likely to be small. Those irregularities with appreciable phasedelay which do show up would be more likely to correspond to saddles of low slope in the phase-frequency characteristic, such as would be caused by a cylindrical or spherical boss on the surface of the planet and convex or concave side slopes of mountains running parallel to the projected axis of rotation on the invisible disk. Due, however, to theirsmall size compared tothe radius of curvature of the planet, their time durationandmagnitude in studyingthe reflection from a rotating planet arelikely to be smallin keeping with Pettengill’s [6] results for the moon. Such reflections, however, will clearly be a t Doppler frequencies corresponding to theirdisplacement from the axis. Further, in distant rotating planets where a weak signal is to be expected, shortterm sums of sampled signals corresponding to the optimum effects of outer limb irregularities noted with their time of occurrence are likely to best emphasize the presence of such physical features, but many irregularities are likely to be missed. Clearly, since the signalfrom the inner Fresnal zones so greatlypredominates, irregularities in the range-time curve may be the best way to obtain evidence of rotation of a remote planet, particularly if the nearest portions of the planet to the observer are regularly repeated in position after a time cycle. E. 0.WILLOUGHBY University of Adelaide Adelaide, South Australia REFEREXES
Fig. 2-Cornu
Eq. (4)becomes exp (-j21.1 X 440)2( V(3.67 X 6 6 0 ) V(3.67 X 220) - 2V(3.67 X 440))
+
confirming the conclusions of (4). Hencethere will belittlecontribution from other than the central Fresnalzones in a perfectly spherical planet. This is in accordance with the J.P.L. experimental results published by Victor and Stevens [4], and Goldstein for the planet Venus [3], [SI.
spiral.
THEPERFECI SPHERE AND THE PRACTICAL CASE The analysis of the perfect sphere helps explain the intense localization of the reflection at the frontedge of the planet. ClearlynearthemainFresnal zones a concentricelevationor depression of the spherical surfaceis likely to have littleeffect on themagnitude of the receivedsignal; however, with an operating wavelength short compared to the surface irregularities,
[l] E.Jahnke, F. Emdeand F. Losch, “Tables of Higher Functions.’ McGran-Hill Book Company, Inc.. New York U. Y. 6th ed. p. 29. 1960. [Z] T.Pearcey,“Table ’ i f tieFresialIktepral,” Commonwealth Scientific and Industrial Research Organization, Melbourne. Australia; 1956. [3] R. M. Goldstein. “Radar Exploration of Venus.” Jet Propulsion Lab., California Institute of Technology, Pasadena, Te:h. Rept. S o . 32-280; May. 1962. [4] W . K. Victor and R. S;evens.‘The1961 JPL Venus radar experiment, IRE TRANS.ON SPACE ELECTRONICS ASD TELEMETRY, V O ~ . SET-8, pp. 84-97; June, 1962; [SI R.M. Goldstein. A technique forthe measurement of the power spectra of very weak signals,” IRE TRANS.ON SPACEELECIRONICS ASD TELEMETRY. vol. SET-8,upp. 170-173; June, 1962. [6] G. H. Pettengill.Measurement 0; lunar reflectivity using the Millstone radar, PROC. IRE ( C o n c s p p c n c c ) . vol. 48, pp. 933-934; May, 1960. [7] $vans,Radioechostudies of the moon.” in Physics and Astronomy of the Moon,” 2. Kopal, Ed.. Academic Press. Inc., New York. N. Y., ch. “11 -211.
(81 R. L.Leadabrand, R. B. Dyce, A . yredriksen. R. I. Pressnell and J. C. Schlobohm,Radio fre; quency scattering from the surface ol the moon, PROC.IRE (Correspondence), vol. 48. DP. 932-933; May, 1960.
