Limb brightening on Uranus: The visible spectrum, II

ZCARUS34, 355-373 (1978)

Limb Brightening on Uranus: The Visible Spectrum, II M I C H A E L J. P R I C E Planetary Science Institute, 2030 E. Speedway Blvd., Suite 201, Tucson, Arizona 85719 AND

O T T O G. F R A N Z Lowell Observatory, P. O. Box 1269, Flagstaff, Arizona 86002

Received May 31, 1977; revised August 29, 1977 New narrow-band (100 X) photoelectric area-scanning photometry of the Uranus disk is reported. Observations were concentrated on the two strong CH4 bands at X 6190 and 7300 X. Adjacent continuum regions at h 6400 and 7500 ~_ were also measured for comparison. Both slit and pinhole scans were made in orthogonal directions. Disk structure in each waveband is apparent through lack of circular symmetry in the intensity distribution over the Uranus image. Polar brightening is especially prominent in the ix 7500-~_waveband. Coarse quantitative determinations of the true intensity distribution over the Uranus disk were made. For the ix 6190-~_ CH4 band, Uranus exhibits a disk of essentially uniform intensity except for a hint of polar brightening. For the ix 7300-~_ CH4 band, moderate limb brightening is apparent. Specifically, the true intensities at the center and limb of the planetary disk are approximately in the proportion 1:2. Extreme limb brightening, with a corresponding intensity ratio greater than 1 : 4, is not permitted by the observational data. 1. INTRODUCTION Belton and Vesculus (1975) pointed out t h a t valuable information for investigating the physical structure of the Uranus atmosphere can be obtained from knowledge of the distribution of brightness over the planetary disk. Qualitative and quantitative infrared studies of the intensity profile of the Uranus disk have been reported b y Westphal (1972) and b y Sinton (1972), respectively. I n Paper I, Price and Franz (1976) studied the wavelength variation in the optical appearance of Uranus using multicolored (X 5500-7600 A), narrow-band (100 A), area-scanning photometry. Eight wavebands were selected. During the 1975 Uranus apparition, absolute limb darkening was found in all spectral regions considered

except for the two CH4 bands at X 6190 and 7300 A. For the X 7300-A band, absolute limb brightening with respect to a uniform disk was found. For the X 6190-A band, no definite conclusions could be drawn regarding the absolute nature of limb brightening. Only limb brightening relative to adjacent continuum regions could be demonstrated. If absolute limb brightening did occur in the X 6190-,~ band, it had to be m u c h less pronounced t h a n in the ~ 7300-/~ band. Quantitative estimates of the degree of either limb darkening or brightening in any waveband could not be obtained from the available observational data. Spatial resolution and photometric accuracy were insufficient. F u r t h e r observations of limb brightening 355 0019-1035/78/0342-0355502.00/0 Copyright ~) 1978 by Academic Press, Inc. All rights of reproduction in any form reserved.

356

PRICE AND FRANZ

on Uranus have since been carried out by other investigators. Avis et al. (1977) reported the photographic detection of albedo features in the X 6190-A CH4 band. Image enhancement processing rcsultcd in the discovery of local polar brightening superimposed on weak, symmetrical, absolute limb brightening. Smith (1977), using a CCD camera, confirmed the existence of absolute limb brightening in the X 8900-~. CH4 band. During the 1976 Uranus apparition, we refined our earlier observations of the planet. Spatial resolution and photometric accuracy had been much improved since 1975. Our objective was to quantitatively determine the degree of limb brightening in each of the two CH4 bands of interest. For the ,X7300-A band, initial qualitative results were reported by Franz and Price (1977). Pinhole scans demonstrated directly the existence of both limb and polar brightening. Polar brightening appeared to be present also at adjacent continuum wavelengths. In this paper, we present a detailed analysis of our 1976 observati(mal data. In a subsequent paper, aFt interpretation of our results in terms of the physical structure of the Uranus atmosphere will be given. 2. OBSEt~VATIONS

Using the equipment and technique described in Paper I, we carried out new photoelectric area-scanning photometry ()f the Uranus disk during the 1976 apparition. Measurements were restricted to four narrow-band ( ~ 1 0 0 A) spectral regions, namely the CH4 bands selected with Filter No. 3 (X 6200 ~,) and Filter No. 7 (X 7300 A), and adjacent "continuum" regions studied through Filter No. 4 (X 6400 .~) and Filter No. 8 (X 7500 ,~). Specifications of the filters were noted in Paper I. Since 1975 major improvements had been made to the optical performance of the Perkins reflector. Specifically, the Casse-

grain secondary was refigured t,) redu(,e spherical aberration. Typical point spread functions produced by the atmospheretelescope combination became narrower by a factor of two compared with those previously obtained. Significant improvement in the spatial resolution of the Uranus disk was a direct result. Photometric signalto-noise ratios were also increased by replacing the earlier EMI-9558 (S-20) tube with an I T T F4085 (S-20) photomultiplier. Uranus was scanned with both slit and pinhole apertures. Slit scans were selected to provide the most reliable photometric data for investigating the true intensity distribution over the disk. Pinhole scans were included to enhance the visibility of limb and polar brightening, and to verify the interpretation of the slit-scan data. Characteristic widths of the pinhole and slit were both chosen equal to 100 ~m (0':645 arc). The slit and scan lengths were each 2 mm (12':9 arc). Point spread function data were obtained by slit-scanning images of individual stars located near the planet. Direct and reverse scans were made along two orientations, north south (N-S) or east-west (E-W). For Uraims, orthogonal scans were used to examine the reality ()f features in the disk profiles. For stellar inmges, orthogonal scans were used to verify that telescope guiding errors were insignificant. Table. I gives the 1976 Uranus observing log. Visual guiding was used with no attempt made to correct for image displacement caused by the wavelength dependence of atmospheric refraction. For the maximum wavelength difference (2000 A) between guiding (X 5500 A) and scanning (7500 h), at the largest zenith distance (51 degrees) encountered in our observations, the relative image displacement never exceeded 0':5 arc. For our slit scans, this displacement is of no consequence. Since both the slit and scan lengths ( ~ 1 3 " arc) were much greater than the sum of both the Uranus angular diameter ( ~ 4 " arc) and

L I M B B R I G H T E N I N G ON U R A N U S

357

TABLE I URANUS OBSERVING

Date (UT)

Scan ~ Type

Direction

1976 May 18 1976 June 16

P S

1976 June 17

P

S~ N N *-~S, E ~ W E --~ W

LOG

Filter no.

Sky b

Remarks ¢

Transparency

Seeing

7, 8 3, 4, 7, 8

4 5

2-3 3-4

7, 8

5

2 3

PSF (t Vir) PSF (L Vir and double stars) PSF (~ Vir)

Scan type is either pinhole (P) or slit (S). b Sky transparency and seeing conditions are given on scale 0-5 (i.e., worst-best). All PSF data were taken using slit scanning. Scan directions and filters used were identical with those employed to obtain the Uranus profiles. Angular scales of both the Uranus and stellar profiles were identical for all scans on all nights. Scale calibration was achieved using double stars of known separation. a

the m a x i m u m image d i s p l a c e m e n t enc o u n t e r e d (0'.'5 arc), no light f r o m t h e p l a n e t was lost w h a t e v e r the scan orientation. F u r t h e r m o r e , no distorti(m ()f a n y scan profile could occur. For our pinhole scans, however, the situation is n o t so clear cut. Because U r a n u s was a l w a y s o b s e r v e d near the time of its local m e r i d i a n transit, the N - S pinhole scans were essentially u n a f f e c t e d b y the p h e n o m e n o n of image displacement. But, the red E - W pinhole scans were l o c a t e d up to 0'~5 arc n o r t h of the disk center w h e n the visual image was centered. T h e chord t r a v e r s e d was t h e n ~ 3 % shorter t h a n the disk diameter. F o r the E - W scans, therefore, limb b r i g h t e n i n g will t e n d to be u n d e r e s t i m a t e d ; limb d a r k e n i n g will be overestimated. We shall r e t u r n to this point in our analysis of the pinhole scan data. C o m p o s i t e U r a n u s a n d stellar profiles were p r o d u c e d to increase the effective signal-to-noise ratio in the p h o t o m e t r y . F o r Uranus, on each night ()f observation, c o m p o s i t e profiles in each w a v e b a n d were o b t a i n e d in each of the two cardinal orient a t i o n s ( N - S , E - W ) for the slit a n d pinhole scans individually. T h e point spread function for each night was o b t a i n e d b y s u m m i n g t o g e t h e r all stellar slit scans. All stellar profiles were symnletrical. I n s p e c t i o n

of the individual stellar profiles s h o w e d no v a r i a t i o n with w a v e b a n d , scan orientation, or scan direction. F o r m a t i o n of the entire set of composite profiles is s u m m a r i z e d in Table II. I n d i v i d u a l U r a n u s slit profiles were t y p i c a l l y o b t a i n e d b y i n t e g r a t i n g 50 to 200 1-see scans; pinhole profiles required 200 to 500 1-see scans. I n d i v i d u a l stellar profiles each consisted of 20 1-see scans. Colocation of the individual U r a n u s a n d stellar profiles prior to composite s u m m a tion was achieved t h r o u g h G a u s s i a n curve fitting b y m e a n s <)f the least:squares t e c h n i q u e t() define the centroids of the individual profiles. W h e r e necessary, scans m a d e in opposite directions were m i r r o r e d on their centroids prior t<) c o m p u t e r summati(m. All U r a n u s composite profiles were normalized to an equal a r b i t r a r y i n t e g r a t e d signal not only to facilitate i n t e r w a v e b a n d comparisons, b u t also to expedite c o m p a r i sons with theoretical predictions. I n essence, the p h o t o e l e c t r o n c o u n t s received f r o m the planet for all points in each scan were s u m m e d to derive the total signal. S u c h a n o r m a l i z a t i o n p r o c e d u r e minimizes the influence of individual p h o t o m e t r i c errors within each o b s e r v a t i o n a l scan, a n d eliminates the need to k n o w the effective geo-

358

PRICE AND FRANZ

metrical albedo in each waveband when comparing observation with theory. By comparison, each stellar profile was scaled to an equal arbitrary intensity at its centroid. 3. THE POINT SPREAD FUNCTION Atmospheric turbulence, together with diffuse scattering from the telescope mirrors, produces the observed point spread function. I n Paper I, we showed t h a t a single Gaussian curve provided only a coarse fit to the distribution of intensity within the image of a point source; a double-Gaussian curve (Fig. 1) provides a far better description of the point spread function. Line integration, slit broadening, and normalization of the double-Gaussian shape are discussed in Appendix I. Composite stellar slit-scans obtained for the nights of 1976 M a y 18, June 16, and June 17 are plotted individually ill Fig. 1. Best-fitting theoretical scans, based on the double-Gaussian representation of the point spread function, are shown for comparison. Optimum values of the P S F parameters for each night are tabulated in Fig. 1.

Interestingly enough, the point spread functions on 1976 M a y 18 and June 17 were essentially identical. On 1976 June 16, to demonstrate the ability of the area scanner to resolve the Uranus image b y slit scanning, a set of close visual binary stars was also measured. Results are illustrated ill Fig. 2. Variations in the point spread function, during each night of observation, are of special interest for the interpretation of the Uranus data. For 1976 June 16, quantitative estimates of the variation in the width of the actual point spread function m a y be made by fitting a single Gaussian curve to each of the 29 individual stellar slit scans by means of the least-squares technique. The derived 1/e widths provide a measure of the constancy of the point spread function. Distortion of the actual point spread function introduced b y slit scanning is not significant in the present context. Results show t h a t the individual rms fluctuation in the 1/e width anmunted to 6.0%. For the composite profile, the rms error in its width should therefore a m o u n t to ~ 1 % , in agreement with the observa-

TABLE II FORMATION OF COMPOSITE PROFILES

Date (UT)

1976 May 18

1976 June 16

1976 June 17

Object

Uranus Uranus PSF Uranus Uranus Uranus Uranus Uranus Uranus Uranus Uranus PSF Uranus Uranus PSF

Scan Type

Direction

P P S S S S S S S S S S P P S

N S N-S All N-S N-S N-S N-S E W E-W E-W E-W All E-W E-W All

Filter

7 8 All 3 4 7 8 3 4 7 8 All 7 8 All

Integrated Total scans l-see scans/ integration 2 2 4 5 4 3 4 4 4 6 8 29 3 2 5

500-801 200 20 50 50 100-200 50 50 50 100-200 50 20 500 100 -200 20

Total 1-see scans/ composite 1301 400 80 250 200 400 2O0 200 200 700 400 580 1500 300 100

LIMB BRIGHTENING ON URANUS

I• "::?. :~.:

' SLIT

I

----Z

,

,PSF

>-

m ~

,

OPTIMUM PARAMETERS

~~

"L~ \

rr"

,

F(r) -- A exp [~--~-)+ S exp

I

f\\.

~ r

359

TE CURVED (1A 91 6) A

'~" ,

5-

"~ , ~

Tr

>-

o~

Iou~ 0".775 1".75 /CLAY18,~ It 0.1250"100 05'.8 5 0 2 " . 0 0

Ill JUNE I?

~

B ~

".~. . . . . . . "-'----..-..----.

I

._

I

cr)

7 bJ F-

rr

_z

m

0

I

I

I

2

I

I

5 4 ANGULAR DISTANCE ("ARC)

I

5

6

Fro. I. The composite stellar slit scans for each night of observation. Theoretical point spread function predictions are compared with observation. Optimum curve fitting only is illustrated. The corresponding PSF parameters are tabulated. tional d a t a presented in Fig. 1. I n v e s t i g a tion of v a r i a t i o n in the point spread function during the nights of 1976 M a y 18 and 1976 June 17 produced similar results. Evidently, m e a n theoretical P S F p a r a m eters for each night of observation are known to an accuracy ~ 1 % . 4. URANUS SLIT SCANS

~.1. Modeling Procedures Deriving the true intensity distribution over the Uranus disk in each w a v e b a n d of interest was the objective of the slit-scan analysis. Our a p p r o a c h was first to model b o t h the size and shape of the planet together with the a d o p t e d " t r u e " distribution of intensity over the disk, next to cmph)y tile known point spread function

in a two-dimensional broadening procedure to derive the p l a n e t a r y image smeared b y atmospheric seeing, t h e n to c o m p u t e the profile which would result from slit scanning the image in one dimension, and finally to normalize the slit-scan prediction to p e r m i t comparison with the observed profile. Full details of the m a t h e m a t i c a l formulation of the problem are contained in Appendix II. Using Strat(;scope I I p h o t o g r a p h s of Uranus, Danielson et al. (1972) derived a p l a n e t a r y equatorial radius of 25,900 -4- 300 k m together with an ellipticity of 0.01 4- 0.01. Although stellar occultations m a y provide i m p r o v e d values for b o t h the radius and ellipticity of the planet, we a d o p t the values given b y Danielson et al. as the best available. Gur theoretical predictions assmned Uranus to be a perfect

360

PRICE AND FRANZ

sphere of radius 25,900 km. Circular s y m m e t r y in the distribution of intensity over the disk was also assumed. Distance from the E a r t h to Uranus on 1976 June 16 was taken from the American Ephemeris and Nautical Almanac to be 17.88 AU. The corresponding angular diameter of Uranus (unbroadened) was therefore 3'.'99

determined by the relative intensities at the limb and center of the disk. Although consideration was given to the predictions b y Belton and Price (1973), our choice of the radial intensity function remained largely arbitrary. Between the limb and center of the disk, the true intensity distribution was assumed to follow an elliptical curve. For both limb brightening and darkening, the slope of the function was taken to be zero at the disk center. At the limb, the slope reaches negative infinity for limb darkening, and positive infinity for limb brightening. Six curves,

are.

Simple theoretical distributions of intensity over the Uranus disk were adopted which could be described by a single parameter chosen to encompass a broad range of situations; this parameter was

SLIT

H

1976

JUNE

16

ADS 11485

T

OPH

e

LYR AB

I _#

L_ SCALE

i

# ,

0

E LYR CD

,

i

I

i

I

5 " ARC

I

,

i

I

I0

Fro. 2. Illustration of atmospheric seeing quality on 1976 June 16. Specimen double star scans obtained in the direction of maximum separation are shown. Se'ms were obhdned by integrating over 20 individual 1-sec sweeps.

L I M B B R I G H T E N I N G ON UI~ANUS

I.O

,/UNIFORM DISK

p=O T(r) ~

r FIG. 3. Theoretical intensity distributions ~ver the Uranus disk selected for the analysis. Intensity, I(r), is considered to be a smooth function of radial distance, r, from the center of the disk. Circular symmetry is assumed.

illustrated in Fig. 3, were used in our analysis. Besides the ease of a uniform disk, both limb-darkening (convex) and limbbrightening (concave) intensity distribu= tions, ranging from m<)derate to extreme, were adopted. For limb darkening, the parameter (p) equal to the ratio of the intensities at the limb and center of the disk was sufficient to describe the distribution. For limb brightening, the parameter (q) equal to the ratio of the intensities at the center and limb filled all identical role. Distortions of the slit scans resulting from variations in atmospheric seeing need to be examined before we embark on a detailed comparison of theory with observation. Sample theoretical slit scans were computed for three extreme m()de]s of the radial intensity distribution, namely the uniform disk, extreme limb brightening (q = 0), and extreme limb darkening (p = 0). Each distribution was subjected to smearing by three distinct point spread functions, described by the <)ptimum set of parameters (A, B, z,, a2) listed ill Fig. 1 for 1976 June 16 together with two extreme variants obtained by changing the individual B, ~,, and z2 parameters in unison by 4-5%. Results are illustrated in Fig. 4.

361

Increasing the PSF width reduces the normalized intensities near the center of the slit scans; intensities in the wings are increased. For the composite Uranus slit scans, the effective PSF parameters should be uncertain by only -~1%. Evidently, variations in the seeing will have a negligible effect on the interpretation of the data. Photometric noise will be significantly greater than profile fluctuations resulting from the cumulative effects of seeing fluctuations. Uncertainty in the slit-scan predictions introduced by inaccurate knowledge of the Uranus radius also requires examination. Sample theoretical slit scans were calculated for three distinct radii; the Daniclson et al. value was adopted together with radii differing from the optimum value by ~:5%. Models chosen for the intensity distribution were again a uniform disk, extreme limb brightening ( q - - 0 ) , and extreme limb darkening (p = 0). The point spread function parameters for 1976 ,June 16 were those listed in Fig. 1. Results are illustrated in Fig. 5. Enlarging the plan'.'tary radius reduces the normalized intensities near the center of the slit scans; intensities in the wings are increased. Since the Uranus radius appears to be uncertain by only ~ 1 % , the corresponding error introduced in the slit-scan predictions will have a negligible effect on the interpretation of the data. Photometric noise in the composite Uranus scans will be significantly greater. ~.2. Results

Observed composite Uranus slit scans, in all four wavebands of interest, are compared with theoretical predictions in Figs. 6 through 9. Discrimination between the individual intensity distributions is readily achieved ,lear the center of the planetary image. Small uncertainties both ill the planetary radius and in the point spread function then have their least influence on

362

PRICE AND FRANZ

the interpretation. Investigation of disk structure through detection of gross a s y m metries and local anomalies in the slit scans is a principal objective of our analysis. No a t t e m p t s have therefore been m a d e either to smooth out the residual photometric noise in the composite scans or to

force-fit a curve through each set of observational data. For each waveband, n o r t h south (N-S) and e a s t - w e s t ( E - W ) orthogonal scans were plotted separately. Polar brightening would manifest itself near the center of the N - S scans, and on the westerly segment of the E - W scans. I n f o r m a t i o n

I

1.0 p=O

SLIT

0.5

0

~

1.0

FLAT >,.
z

w I.-Z 1.0

CONCAVE

q=O

o

0 ANGULAR DISTANCE ("ARC)

FIG. 4. Analytical uncertainties introduced by atmospheric seeing fluctuations. Theoretical Uranus slit scans are shown computed for 1976 June 16. The Uranus disk was taken as circularly symmetric with an apparent, angular diameter of 3'.'99 arc. Seeing broadening was computed for a uniformly bright disk, and for the two cases of extreme limb darkening and extreme limb brightening shown in Fig. 3. The optimum PSF parameters tabulated in Fig. 1 were used together with two alternate sets of parameters obtained by varying B, or,, and ¢2 together by :t:5(~v.

LIMB BRIGHTENING ON URANUS

363 ,

i

1.0 CONVEX

p =0

SLIT

0.5

0 A

oo

1.0 FLAT

Z .-~

or n~

~0.5 or

>-

z

="

0

1.0 CONCAVE

q: 0

0 6

5

4

3 I 0 I 2 3 ANGULAR DISTANCE ("ARC)

5

6

FIG. 5. Analytical uncertainties introduced by changes in the Uranus radius. Theoretical Uranus slit scans are shown computed for 1976 June 16. Three models for the radial distribution of intensity over the disk were used. Computations were made for a uniformly bright circular disk, and for the two eases of extreme limb darkening and extreme limb brightening shown in Fig. 3. Seeing broadening was described by the optimum set of PSF parameters tabulated in Fig. 1. The optimum Uranus radius was taken as 25,900 km; ~wo alternative Uranus radii, obtained by varying the optimum value by -4-5%, were also used. g i v e n ill the ExplaTtatory Supplement of the

American Ephemeris and Nautical Alma~ac shows t h a t , oil 1976 J u n e 16, t h e n o r t h pole of U r a n u s was l o c a t e d at p o s i t i o n angle 278.8 degrees a t a d i s t a n c e of 0.69 U r a n u s

r a d i i from the disk center. L o c a t i o n of t h e pole o n t h e disk did n o t c h a n g e signific a n t l y t h r o u g h o u t t h e 1976 o b s e r v i n g SOaSOn.

Data

for the h 6190-•

CH4 b a n d arc

364

PRICE AND FRANZ

p r e s e n t e d i n Fig. 6. B()th N - S a n d E - W scans suggest t h a t t h e o p t i m u m t r u e r a d i a l i n t e n s i t y d i s t r i b u t i o n c o r r e s p o n d s to a u n i f o r m disk. B u t w e a k l i m b d a r k e n i n g (p >_ 0.5) a n d w e a k l i m b b r i g h t e n i n g (q >__ 0.5) are also p e r m i t t e d b y t h e N S a n d E - W scans, r e s p e c t i v e l y . D a t a for t h e a d j a c e n t " c o n t i n u u m " region a t X 6400 A are p l o t t e d i n Fig. 7. L i m b d a r k e n i n g is i

|

i

f

readily apparent in both the N - S and E - W scans. F o r t h e N S scan, t h e t r u e r a d i a l intensity distribution can range from a u n i f o r m disk to w e a k l i m b d a r k e n i n g (p > 0.5). F o r t h e E - W scan, t h e distrib u t i o n c a n r a n g e f r o m m o d e r a t e to e x t r e m e l i m b d a r k e n i n g (0.5 >_ p >_ 0). G i v i n g e q u a l c o n s i d e r a t i o n to b o t h t h e N - S a n d E - W scans, we will a d o p t m o d e r a t e l i m b d a r k e n -

i

!

I

=

1.0

I

|

|

1976 JUNE 16 N-')S

._,0.5 Iz

>.. p.

=

~

0

-~,

< 1.0 >.. I---

-

-:-

E-)W

I.u I--7 0.5

I

6

4

I

I

I

I

3 2 I 0 I 2 3 ANGULAR DISTANCE ("ARC)

I

I

4

6

~'IG. 6. Composite Uranus slit scans obtained on 1976 June 16 for the X 6200-~_ waveband. Theoretical slit scans are compared with observation. The Uranus disk was taken as circularly symmetric with an apparent angular diameter of 3'!99 arc. Seeing broadening was described by the optimum set of PSF parameters tabulated in Fig. 1. All six models for the radial distribution of intensity over the Uranus disk shown ill Fig. 3 were used. All theoretical and observational scans were normalized to a fixed arbitrary flux from the playlet. In the abscissa, the zero point corresponds to the centroid of each observed and theoretical scan profile. Centroids of the observed scans will coincide with the physical center of the disk only if the intensity distribution is circularly symmetric. For information purposes, the observable pole of Uranus is located 1'.~4arc west, 0'~ 2 arc north of the physical center of the disk.

365

LIMB BRIGHTENING ON URANUS I

I

!

I

I

I

I

I

I

I

I

I

1.0

N.-)S

.

A)

..0.5 or} I-)n-

0

n,.

.~ 1.0

E-)W

>.. I-

co z

IJJ Iz

0.5

0

,

,

-.

i

I

I

6

5

4

I

I

I

I

i

3 2 I 0 I ANGULAR DISTANCE

I

I

2 3 ("ARC)

_ -

I

I

i

4

5

6

FIo. 7. Composite Uranus slit scans obtained on 1976 June 16 for the X6400-~ waveband. Theoretical slit scans are compared with observation. The Uranus disk was taken as circularly symmetric with an apparent angular diameter of 3':99 are. Seeing broadening was described by the optimum set of PSF parameters tabulated in Fig. 1. All six models for the radial distribution of intensity over the Uranus disk shown in Fig. 3 were used. All theoretical and observational scans were normalized to a fixed arbitrary flux from the planet. In the abscissa, the zero point corresponds to the eentroid of each observed and theoretical scan profile. Centroids of the observed scans will coincide with the physical center of the disk on,ly if the intensity distribution is circularly symmetric. For information purposes, the observable pole of Uranus is located 1':4 are west., 0'.'2 are north of the physical center of the disk. i n g (p = 0.5) as the o p t i m u m fit to t h e X 6400-• d a t a . D a t a for t h e s t r o n g X 7300-fi. CH4 b a n d are p r e s e n t e d i n Fig. 8. I n spite of t h e r e s i d u a l p h o t o m e t r i c errors i n b o t h N - S a n d E - W c o m p o s i t e scans, l i m b b r i g h t e n i n g is e v i d e n t . Precise d e t e r m i n a t i o n of its m a g n i t u d e is difficult however. F o r b o t h t h e N - S a n d E - W scans, t h e t r u e r a d i a l

intensity distribution can range from a u n i f o r m disk to s u b s t a n t i a l l i m b b r i g h t e n ing (q = 0.25). B u t e x t r e m e l i m b b r i g h t e n ing (q = 0) c a n n o t be r e c o n c i l e d w i t h t h e observational data. Giving equal considerat i o n to t h e N S a n d E - W scans, we will a d o p t m o d e r a t e l i m b b r i g h t e n i n g (q = 0.5) as t h e o p t i m u m fit to the X 7300-A d a t a . Observations in the adjacent "continuum"

366

PRICE AND FRANZ

region at X 7500 ~. are ph~tted in Fig. 9. For the N S scan, t h e t r u e r a d i a l i n t e n s i t y d i s t r i b u t i o n c a n range from a uniform disk to moderate l i n d ) brightening (q = 0.5). For the E W scan, t h e d i s t r i b u t i o n can r a n g e from a uniform disk to moderate l i m b darkening (p = 0.5). Evaluating b o t h s c a n d i r c c t i o n s t o g e t h e r , o n e c a n conclude t h a t on average t h e d i s t r i b u t i o n of intensity

I

!

I

|

in the h 75()0-A 1)ands c()rresp()l~ds at)pr()ximately t<) a uniform disk. For the ~ 6400-, 7300-, and 75()0-A wavebands, striking differences are a p p a r e n t between the N S and E W scans. The E - W scans exhibit a distinctly greater i n t e n s i t y near t h e c e n t c r of t h e p l a n e t a r y image. S i n c e N - S and E - W s c a n s are n o r m a l i z e d to an identical total signal, the E:-W scans

I

I

I

i

I

I.O

I

I

|

1976 JUNE 16

N-)S

500A)

0.5 (t) z ::D >.nr fY

i-

~

.

.

.

.

..

0 1.0

•

Z

0.5

0 i

I

I

I

6

5

4.

3

I

I

I

I

I

|

2 I 0 I 2 3 ANGULAR DISTANCE ("ARC)

I

I

I

4

5

6

FIG. 8. Composite Uranus slit scans obtained oil 1976 June 16 for the X 7300-.& wavcband. Theoretical slit scans are compared with observation. The Uranus disk was taken as circularly symmetric with an apparent angular diameter of 3'.'99 arc. Seeing broadening was described by the optimum set of PSF parameters tabulated in Fig. 1. All six models for the radial distribution of intensity over the Uranus disk shown in Fig. 3 were used, All theoretical and observational scans were normalized to a fixed arbitrary flux from the planet. In the abscissa, the zero point corresponds to the centroid of each observed and theoretical scan profile. Centroids of the observed scans will coincide with the physical center of the disk only if the intensity distribution is circularly symmetric. For information purposes, the observable pole of Uranus is located 1'!4 arc west, 0':2 arc north of the physical center of the disk.

LIMB BRIGHTENING ON URANUS !

!

!

!

!

I

I

I

1.0

367 !

!

!

1976 JUNE 16

0.5 Z

~

m

0

~:1.0 ).z b.I I.--

Z-O. ~

I

I

I

I

I

I

I

I

I

I

I

I

I

6

5

4

3

2

I

0

I

2

3

4

5

6

ANGULAR DISTANCE ("ARC) FIG. 9. C o m p o s i t e U r a n u s slit s c a n s o b t a i n e d on 1976 J u n e 16 for t h e X 7500-~ w a v e b a n d . T h e o retical slit s c a n s are c o m p a r e d w i t h observation. T h e U r a n u s disk was t a k e n as circularly s y m m e t r i c w i t h a n a p p a r e n t a n g u l a r d i a m e t e r of 3'.'99 arc. Seeing b r o a d e n i n g was described b y t h e

optimum set of PSF parameters tabulated in Fig. 1. All six models for the radial distribution of intensity over the Uranus disk shown in Fig. 3 were used. All theoretical and observational scans were normalized to a fixed arbitrary flux from the planet. In the abscissa, the zero point corresponds to the centroid of each observed and theoretical scan profile. Centroids of the observed scans will coincide with the physical center of the disk only if the intensity distribution is circularly symmetric. For information purposes, the observable pole of Uranus is located 1'.'4 arc west, 0':2 arc north of the physical center of the disk. m u s t also h a v e a s l i g h t l y n a r r o w e r profile t h a n t h e N - S scans. C o n s i d e r i n g t h e s c a n g e o m e t r y , one m i g h t c o n c l u d e t h a t U r a n u s is s i g n i f i c a n t l y o b l a t e . B u t a n e l l i p t i c i t y ~-0.1 w o u l d b e r e q u i r e d to p r o d u c e t h e effect. S t u d i e s b y D a n i e l s o n et al. (1972) a p p e a r to p r e c l u d e t h a t i n t e r p r e t a t i o n . I f U r a n u s is in f a c t e s s e n t i a l l y s p h e r i c a l , one is o b l i g e d to c o n c l u d e t h a t t h e d i s t r i b u t i o n

of i n t e n s i t y o v e r t h e d i s k is n o t c i r c u l a r l y symmetric. Disk structure must be present in e a c h of t h e a b o v e w a v e b a n d s . F o r t h e X 6190-A CH4 b a n d , a s l i g h t a s y m m e t r y in t h e E - W s c a n s u g g e s t s t h e p r e s e n c e of w e a k p o l a r b r i g h t e n i n g . One final p o i n t m a y b e m a d e . I n all f o u r wavebands, the N-S scans show a slight a s y m m e t r y in t h e w i n g s ; t h e n o r t h e r l y

368

P R I C E AND F R A N Z

s e g m e n t is d e f i n i t e l y b r i g h t e r t h a n t h e southerly segment. One might be tempted t o i n t e r p r e t t h i s a s y m m e t r y in t e r m s of t h e recently discovered Uranus ring system. But such an interpretation

would be highly

speculative.

1

1

!

I

I

I

5, UIIANUS PINltOIA,3 SCANS Our slit-scan results were confirmed by the pinhole data. Interpretation of t h e p i n h o l e s e a n s w a s c a r r i e d o u t in a m a n n e r e s s e n t i a l l y i d e n t i e a l t o t h a t e m p l o y e d for t h e slit scans. N o e h a n g e s w e r e m a d e in t h e

I

I

I

I

1

I

I

1976 MAY 18 PINHOLE SCANS (~.7300A)

1.0 N-~S

.\

0,5

>Qc ¢r ~-

0

.......-

. -.

. ~.-:"

• t."

• .-.,.:

rr 1.0

E~W

1976 JUNE 17

>Fz w z --0.5

////' 0

•% I

6

•

~ "1

5

.-...

° I

4

•

• I

I

I

I

I

3 2 I 0 I A N G U L A R DISTANCE

~,A •

•

,p

I

I

,•o.

I

;

I

".'~ I

2

:3

4

5

6

("ARC)

FIG. 10. Composite Uranus pinhole scans obtained on 1976 May 18 and on 1976 June 17 for the X 7300-~ waveband. Theoretical pinhole scans through the image center are compared with observation. The Uranus disk was taken as circularly symmetric with a mean apparent angular diameter of 4'!03 are. Seeing broadening was described by the optimum set of PSF parameters tabulated in Fig. 1• All six models for the radial distribution of intensity over the Uranus disk shown in Fig. 3 were used. All theoretical and observational scans were normalized to a fixed arbitrary flux from the planet. In the abscissa, the zero point corresponds to the eentroid of each observed and theoretical scan profile. Centreids of the observed scans will coincide with the physical center of the disk only if the intensity distribution is circularly symmetric. Moreover, the scans must be precisely across a diameter of the planetary disk. For information purposes, the observable pole of Uranus is located 1':4 arc west, 0'!2 are north of the physical center of the disk.

LIMB i

,

,

BRIGHTENING ,

=

1

369

ON URANUS i

,

r

I.O

,

,

|

|

1976 MAY 18 PINHOLE SCANS ( X 7 5 0 0 A )

N )S

0.5-

I-

0

m

n," 1.0

".j.. E-)W

>_

;

• .-" "" ~."

1976 JUNE 17 PINHOLE SCANS ( h 7 5 0 0 A )

O3 Z LLI I--

_z

05

O

. I

6

•

,.~. I

5

.%..

. I

4

.o I

:3

I

2

I

I

I

0

I

I

ANGULAR DISTANCE

I

I

2 5 ('*ARC)

I

4

I

5

6

FIG. 11. Composite Uranus pinhole scans obtained on 1976 May 18 and 1976 June 17 for the ~, 7500-~ waveband. Theoretical pinhole scans through the image center are compared with observation. The Uranus disk was taken as circularly symmetric with a mean apparent angular diameter of 4'.'03 arc. Seeing broadening was described by the optimum set of PSF parameters tabulated in Fig. 1. All six models for the radial distribution of intensity over the Uranus disk shown in Fig. 3 were used. All theoretical and observational scans were normalized to a fixed arbitrary flux from the planet. In the abscissa, the zero point corresponds to the centroid of each observed and theoretical scan profile. Centroids of the observed scans will coincide with the physical center of the disk only if the intensity distribution is circularly symmetric. Moreover the scans must be precisely across a diameter of the planetary disk. For information purposes, the observable pole of Uranus is located 1':4 arc west, 0'.'2 arc north of the physical center of the disk. U r a n u s disk model. O n l y d i a m e t r i c s c a n n i n g of t h e U r a n u s i m a g e was considered. U n c e r t a i n t i e s b o t h i n t h e a t m o s p h e r i c seeing a n d in t h e p l a n e t a r y r a d i u s were insignific a n t w h e n c o n s i d e r e d i n t h e c o n t e x t of t h e p h o t o m e t r i c noise r e m a i n i n g i n t h e composite p i n h o l e scans. M o d i f i c a t i o n s to t h e theoretical formulation introduced by

c h a n g i n g the s c a n n i n g a p e r t u r e f r o m a slit t o a p i n h o l e are discussed i n A p p e n d i x I I . T h e o r e t i c a l p i n h o l e scans were c a l c u l a t e d for t h e six r a d i a l i n t e n s i t y d i s t r i b u t i o n s i l l u s t r a t e d i n Fig. 3. A t m o s p h e r i c seeing was described b y t h e o p t i m u m set of P S F p a r a m e t e r s (A, B, ~1, ~2) listed i n Fig. 1 for b o t h t h e 1976 M a y 18 a n d 1976 J u n e 17

370

PRICE AND FRANZ

observations. For the two nights in question, a mean E a r t h - U r a n u s distance of 17.74 AU was obtained from the American Ephemeris and Nautical Almanac. The corresponding mean angular diameter (unbroadened) of Uranus was 4'.~03 arc. Adopting a mean diameter introduces an error of less t h a n 1% in the value for each night. Observed composite Uranus pinhole scans, for the ~ 7300 and 7500-fk wavebands, are compared with theoretical predictions in Figs. 10 and 11, respectively. For each waveband, n o r t h - s o u t h (N-S) and east-west (E-S) scans are plotted separately to investigate the presence of disk structure. Compared with the slit-scan observations, significantly greater photometric noise remains in the pinhole data. But, fortuitously, the theoretical pinhole scans exhibit a far greater sensitivity to the shape of the true intensity distribution. Note, however, t h a t telescope guiding errors (rms deviation -~0'.~2 arc) are potentially more serious for pinhole scans t h a n for slit scans. Errors in guiding can cause the observed pinhole scans to exhibit rather less limb brightening, and rather more limb darkening t h a n is actually present in the Uranus image. Earlier, we pointed out that, for the E - W pinhole scans only, an identical effect is introduced b y image displacement ( < 0 ' : 5 arc) resulting from atmospheric dispersion. For the pinhole observations, the observable pole of Uranus lies 1"4 arc west, 0('2 arc north of the center of the planetary disk. Northerly displacement of the E - W pinhole scans, with respect to the pole, will therefore a m o u n t to less than 0'.'3 arc. Since our pinhole diameter was 0'.'645, the pole will always be included in the E - W scans, but will always be excluded from the N S scans. Differences in the visibility of polar brightening should therefore be apparent between the E - W and N - S pinhole scans. For the X 7300-/~ CH4 band, Fig. 10 shows that polar brightening is present in the E - W scan and, as expected, is not

in the N - S scan. Using the N S scan to derive limits to the shape of the true intensity distribution, we conclude t h a t limb brightening is definitely present in this waveband. Extreme possibilities range from a uniform disk to moderate limb brightening (q = 0.25). Extreme limb brightening (q -- 0) is excluded. The best match between observation and theory is achieved for moderate limb brightening (q = 0.5). Evaluating the pinhole and slit-scan data together, we estimate t h a t the maximum permitted degree of limb brightening corresponds to q = 0.25. For the ~ 7500-/~ data, Fig. 11 shows t h a t polar brightening is present in the E - W scan, but not in the N - S scan. Evidently, polar brightening must be highly locMized on the Uranus disk. Using the N S scan to derive limits to the shape of the true intensity distribution, we estimate t h a t limb darkening may be present in this waveband. Possibilities range from weak limb brightening (q = 0.5) to extreme limb darkening. The best match between observation and theory appears to lie between a uniform disk and moderate limb darkening ( p - - 0 . 5 ) . For the E - W scan, polar brightening is so significant t h a t it grossly affects the overall shape of the profile. Possibilities for the true intensity distribution appear to range fr(~m a uniform disk to extreme limb darkening (p = 0). Even the latter distribution is not sufficiently extreme to completely encompass the polar brightening. Considering the pinhole and slit-scan results together, we conclude t h a t the basic distribution of intensity in this waveband corresponds to an essentially uniform disk upon which is superimposed significant polar brightening. 6. CONCLUSIONS AND DISCUSSION Coarse quantitative information on the true radial intensity distribution over the Uranus disk has been derived in four

LIMB BRIGHTENING ON URANUS selected wavebands. Lack of circular symm e t r y in the intensity distributions indicates the presence of disk structure, especially polar brightening, in each waveband. For the X 6190-A CH4 band, the distribution corresponds to a uniform disk upon which a hint of polar brightening is superimposed. For the X 6400-A region, moderate limb darkening (p = 0.5) was found. For the X 7300-A CH4 band, moderate limb brightening (q = 0.5) was discovered. Extreme limb brightening (q < 0.25) was not permitted b y the observational data. For the X 7500-A waveband, the distribution corresponds to a uniform disk upon which significant localized polar brightening is superimposed. While the X 7300-A CH4 band observations exhibit absolute limb brightening with respect to a uniform disk, the X 6190-A CH4 band data show only relative limb brightening with respect to nearby continuum regi(ms. Our results m a y be compared with those obtained by Sinton (1972) for the X 8900-A CH4 band. Both limb and polar brightening were found. Sinton fitted his observations with a radial intensity distribution consisting of a uniform disk combined with limb brightening proportional to 1/~, where /z equals the cosine of the angle made by the incident/emergent ray with respect to the local outward normal to the atmosphere. Both components of the intensity were taken to contribute equally at the center of the Uranus disk. In the notation of Appendix II, the limb brightening component takes the functional form (1 -- r2) - 1 / 2 , 0

< r < 1.

Our coarse analytical technique used in this paper to explore the information content of the photometric scans of Uranus has several basic limitations. First, the assumption of smooth elliptical distributions of radial intensity over the disk is entirely arbitrary. Second, the use of circular s y m m e t r y prevents investigation ()f azimuthal structure on the disk. In fact, direct deeonvolution of the Uranus scans

371

is required to thoroughly investigate the two-dimensional photometric structure of its disk. In principle, Fourier analytical techniques, operating on the Uranus slit scans made in multiple directions, can be used. But practical application of Fourier analysis to the Uranus images remains to be demonstrated. APPENDIX I: THE POINT SPREAD FUNCTION Our empirical studies show t h a t the two-dimensional (x, y) image of an astronomical point source formed b y the Flagstaff atmosphere-Perkins telescope combination can be accurately described b y a circularly symmetric normalized intensity profile given b y F (r) = [-~r(A a 2 + Ba22) I-lEA exp ( -- r2/a 12) + B exp(--r2/a2~)-], (1.1) where

r = (x: + y~)t/2

(1.2)

and A, B, zl, and a2 are constants for the image under consideration. Next, consider a slit of infinite length, and infinitesimal width, scanning the twodimensional image in one dimension (x). Line integration of the point spread function produces a profile given b y

L(z) =

L

F(z, y)(~y,

(1.3)

which reduces to

L(x)

= [ - w l / ~ ( A q i 2 + Bq22)-] -1

× [A~I e x p ( - x V ~ l ~)

+ Bz2 exp(--x2/a22)].

(1.4)

For a slit of finite width, A, the lineintegrated profile is broadened to produce S(x) = (l/A)

L(r)d~,

(1.5)

where the integration limits are a = x - - A/2,

b = x + A/2.

(1.6)

372

PRICE A N D FRANZ

The slit-scan profile of the point spread function reduces to

and

S(x)

Equation (2.6) reduces to

= [ - 2 A ( A ~ 2 + B~2'~)J -1

X {Aa~2[erf(b/aO -- e r f ( a / a d 3 -t- Ba22[-erf (b/a2) -- erf (a/a~) ~ }. (1.7) Optimum values for the parameters A, B, a~, a2 are obtained by direct comparison of theoretical predictions with the observational data.

p~= r 2 + u 2 -- 2ru cos 0.

I ( u ) = E2/(Aa~ 2 + Bz2~)~(T1 + T2), (2.9) where F

1

5/'1 = A exp(-- u2/z12)/ D(r)r J0

X Io(2ru/~rl '2) exp(--r2/z12)dr A P P E N D I X II: THE URANUS IMAGE

1. True disk profiles. The Uranus disk is taken to be circular and of unit radius. Its true intensity profiles, D(r), were chosen to encompass the possibilities of extremes in b o t h limb darkening and limb brightening. Limb darkening was described b y D(r) = p + (1 -- p)(1 - /.2)1/2, 0<

r<

1,

(2.1)

(2.8)

2/2 = B exp(--u2/~r22

,,

)fl

(2.1o)

D(r)r

dO

X I o ( 2 r u / ~ 2) exp(-- r2/~22)dr. and I0 denotes the modified Bessel function. 8. Line-integration and slit-broadenir~g. Consider a slit of infinite length, and infinitesimal width, scanning the Uranus image in one dimension (x). Line integration produces

where p = D(1)/D(O).

(2.2)

L(x) =

Limb brightening was described b y D(r) = 1 -- (1 -- q)(1 -- r2) '/2, 0 < r < 1, where q = D(O)/D(1).

0 < r < 1.

(2.3) (2.4)

S(:~) = (1/~)

///0

D(r)F(p)rdrdO,

(2.5)

(2.12)

where the integral limits are given by a = x-

A/2,

b = x+

A/2.

(2.13)

4. Pinhole i~degration. Our pinhole ealculati(ms consider only diametric scans across the Uranus image. For a pinhole of radius rl, numerical integration over the pinhole aperture is required to obtain the pinhole scan profile, given by

(2.6) P(x) =

where the radial distance, u, from the center of the image is given by u = (x 2 + y2)U~

L(~-)dL ~ a

2. Seeing broadening. E v e r y element of the Uranus disk will be affected b y seeing broadening. Smearing b y the point spread function will produce a circularly symmetric intensity~profile of the planetary image given by I(u) =

(2.11)

For a slit of finite width, A, the lineintegrated profile is broadened to produce a scan profile given b y

The uniform disk is described by D(r) -- 1,

I(x, y)dy. ~c

(2.7)

I(~, 6)~(1~d6,

(2.14)

where I ( , , ¢~) ~ I (u)

(2.15)

L I M B B R I G H T E N I N G ON U R A N U S with u ~ = x 2 + ~2 + 2x~ cos ¢~.

(2.16)

ACKNOWLEDGMENTS This research was supported by the National Aeronautics and Space Administration under contract NASW-2983, and with the support of grants from the National Science Foundation. Some of the data processing was carried out with the support of NASA Grant NGR-03-003-001. The U.S. Naval Observatory, Flagstaff Station, generously lent us an I T T F4085 photonmltiplier for the project. This is Planetary Science Institute Contribution No. 71. REFERENCES

AvIs, C. A., SMITH, H. J., BERGSTRALH,J. T., AND SANDMANN, W. H. (1977). Photometric determination of the rotation period of Uranus. Paper presented at the Eighth Annum Meeting of the American Astronomical Society/Division for Planetary Sciences, Honolulu, Hawaii, 1977 January 19-22. BELTON, M. J. S., AND PRICE, M. J. (1973). Limb-

373

brightening on Uranus: A prediction. Astrophys. J. 179, 965-970. BELTON, M. J. S., AND VESCULUS, F. E. (1975). Why image Uranus? Icarus 24, 299-310. I)ANIELSON~ R. E., TOMASKO,M. G., AND SAVAGE, B. D. (1972). High resolution imagery of Uranus obtained by Stratoscope II. Astrophys. J. 178, 887-900. FRANZ, O. G., AND PainE, M. J. (1977). Uranus: Limb and polar brightening at ), 7300 3~. Astrophys. J. 214, L145--L146. PRICE, M. J., AND FRANZ, O. G. (1976). Limbbrightening on Uranus: The visible spectrum. Icarus 29, 125-136. SINTON, W. M. (1972). Limb and polar brightening of Uranus at 8870 ~_. Astrophys. J. 175, L131L133. S~ITH, B. A. (1977). Uranus photography in the 890-nm absorption band of methane. Paper presented at the Eighth Annum Meeting of the American Astronomical Society/Division for Planetary Sciences, Honolulu, Hawaii, 1977 January 19-22. WESTPHAL, J. A. (1972). Comment at the Third Annual Meeting of the American Astronomical Society/Division for Planetary Sciences, Kona, Hawaii, 1972 March 21-24.