Neptune's rotation period: A correction and a speculation on the difference between photometric and spectroscopic results

ICARUS 42, 71-78 (1980)

Neptune's Rotation P e r i o d A Correction and a Speculation on the Difference between Photometric and Spectroscopic Results M I C H A E L J. S. B E L T O N , L. W A L L A C E , AND S E T H A N N E H. H A Y E S Kitt P e a k National Observatory ~ Tucson, Arizona 85726

AND

M I C H A E L J. PRICE Science Applications, Inc., Tucson, Arizona 85726 Received October 17, 1979; revised J a n u a r y 2, 1980 An error in the H a y e s and Belton (1977), Icarus 32, 383-401) estimate o f the rotation period of N e p t u n e is corrected. If N e p t u n e exhibits the same degree o f limb darkening as U r a n u s near 4900 ]k, the rotation period is 15.4 _+ 3 hr. This value is compatible with a recent spectroscopic determination of M u n c h and Hippelein (1979) w h o find a period of 11.2+[:~ hr. However, if, as indirect evidence suggests, the law o f darkening on N e p t u n e at these w a v e l e n g t h s is less p r o n o u n c e d than on U r a n u s , then the above e s t i m a t e s m a y need to be lengthened by several hours. Recent photometric data are independently analyzed and are found to admit several possible periods, none of which can be confidently a s s u m e d to be correct. T h e period of N e p t u n e m o s t probably falls s o m e w h e r e in the range 15-20 hr. T h e H a y e s - B e l t o n estimate o f the period of U r a n u s is essentially unaffected by the a b o v e - m e n t i o n e d error and remains at 24 +_ 4 hr. All o b s e r v e r s agree that the rotation period of U r a n u s is longer than that of N e p t u n e .

agreement between this value and their own estimate o f 15.0 +4:0 hr. We have reexamined the procedures by which the original H a y e s - B e l t o n seeing corrections were computed and have, in fact, located two serious sources of error. These are the failure to simulate properly the procedures by which the spectroscopic data were digitized, and a poor choice of integration limits in the seeing quadrature. The first o f these errors leads to a systematically increasing underestimate of the magnitude of the seeing correction factor as the seeing condition gets worse. This is the behavior pointed out by Munch and Hippelein. The second error introduces " n o i s e " into the seeing correction factors as a result of varying displacements o f the pattern of quadrature points in the seeing algorithm with respect to the edge of the planetary disk. These errors have been corrected and the

I. E R R O R I D E N T I F I C A T I O N

In a recent paper Munch and Hippelein (1979) h a v e shown that the correction factors used by Hayes and Belton (1977) to remove the effects of atmospheric seeing from measurements of spectroscopic line tilts deviate strongly from analytical predictions. T h e y find that this deviation systematically increases as the seeing conditions deteriorate. They suggested that a correct appraisal of the H a y e s - B e l t o n Neptune data should probably yield a period near 15 ___3 hr (rather than the published 22 _+ 4 hr) in agreement with their own independent measurement of 11.2_+1:4 ~ hr. The H a y e s Belton estimate of Uranus' period (24 ___ 3 hr) was not anticipated to be affected by this error, but they nevertheless note disO p e r a t e d by the Association o f Universities for R e s e a r c h in A s t r o n o m y , Inc., u n d e r contract with the National Science Foundation. 71

0019-1035/80/040071-08502.00/0 Copyright ~ 1980by Academic Press, Inc. All rights of reproduction in any form reserved.

72

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% 'o'2' o',' o'8' o'.8' ,'o' ,'2 FIG. 1. C o m p a r i s o n o f s e e i n g c o r r e c t i o n f a c t o r s c o m p u t e d by M u n c h a n d H i p p e l e i n (1979) a n d t h o s e c o m p u t e d b y the c o r r e c t e d H a y e s - B e l t o n p r o g r a m . In this figure M H r e f e r s to M u n c h - H i p p e l e i n a n d B G to B r o w n - G o o d y (1977). tr is the s e e i n g p a r a m e t e r a n d M l ( r ) is t h e m e a n d i s p l a c e m e n t o f a s p e c t r a l line m e a s u r e d at a d i s t a n c e r f r o m the c e n t e r o f the disk. In this g r a p h the line d i s p l a c e m e n t at the l i m b is n o r m a l ized to r and the s e e i n g p a r a m e t e r is n o r m a l i z e d to the p l a n e t a r y r a d i u s [cf. M u n c h a n d H i p p e l e i n (1979)l. T h e n u m b e r s w h i c h are a t t a c h e d to the v a r i o u s c u r v e s in t h e d i a g r a m r e f e r to the p o s i t i o n o u t f r o m t h e c e n t e r o f the d i s k (in units o f the p l a n e t a r y r a d i u s ) at w h i c h the s e e i n g c o r r e c t i o n f a c t o r s are e s t i m a t e d .

original H a y e s - B e l t o n data reanalyzed. If we assume that the law of darkening on the two planets is identical to that derived by Danielson et al. (1972) from Stratoscope II observations of Uranus, we find the following values for the rotation periods of the two planets:

approximated (to about 6% of the correct value) by the analytical results in Fig. 1. Table III gives the results of the present analysis and can be directly compared with the results originally published by Hayes and Belton (1977) in their Table VI. II. D I S C U S S I O N

(a) The period of Neptune Table I displays the most recent estimates of N e p t u n e ' s rotation period and reveals a disturbingly wide range of values. The results reported in this paper are compatible not only with the rapid rotation found by Munch and Hippelein, but also with the much longer period of 18.44 hr favored by Slavsky and Smith (1978). The intercomparison of the latter two apparently mutually exclusive determinations is complicated by the existence o f a possible period in the Slavsky and Smith data near 11 hr that was originally rejected by them on the grounds that this period produces an implausible light curve. This rejection is now known to have been premature and Smith and Slavsky (1979) reported finding reasonably light curves for assumed periods near 10.42 and 18.44 hr. The values of the dynamical and optical TABLE I

Uranus: Neptune:

24.4 +_ 4 hr, 15.4 +_ 3 hr.

The conclusions of H a y e s - B e l t o n on the orientation of the polar axes and sense of spin of the two planets are unchanged. As an illustration of the correctness of the seeing algorithm, Fig. 1 shows the agreement between seeing corrections evaluated by our program and those calculated by Munch and Hippelein. In this comparison we employ the same definition for the " c e n t e r " of a spectroscopic line as was used by them. However, the actual correction factors used in the analysis depend on how the line center and tilt (cf. Hayes and Belton, 1977) of a line is defined in the data reduction process, and they are only roughly

RECENT SPECTROSCOPIC AND PHOTOMETRIC ESTIMATES OF THE ROTATIONAL PERIODS OF NEPTUNE

AND

URANUS

Source Neptune 1. This paper 2. Munch and Hippelein (1979) 3. Slavsky and Smith (1978) 4. Cruikshank (1978) Uranus 1. Hayes and Belton (1977) 2. Munch and Hippelein (1979) 3. Brown and Goody (1979) 4. Trauger et al. (1978) 5. Trafton (1977) Method I Method II 6. Smith and Slavsky (1979, abstract) 7. Dunham and Elliot (1979, abstract)

Period (hr)

15.4

± 3 11.2+~:~ 18.44 -+ 0.01 19.53 -+ 0.05 18.17 -+ 0.05 24

-+ 3 15.0+~:° 16.16 -+ 0.33 13.0 -+ 1.3

> 15.4 23+~ 23.923 + 0.003 12.8

-+ 1.7

PLANETARY ROTATION oblateness of the planet can in principle be used to discriminate b e t w e e n e x t r e m e values of the rotation period (cf. H a y e s and Belton, 1977), however, in the case of N e p tune e v e n the best values of these parameters ( K o v a l e v s k y and Link, 1969; Freeman and Lynga, 1970) do not have sufficient accuracy. The o p t i m u m values of oblateness found by the a b o v e authors indicate a period near 14 hr, but the formal standard deviations are large enough to easily allow periods as short as 11 hr and as long as 18 hr. In order to arrive at a position on the most likely value of N e p t u n e ' s period, we have made an independent analysis of the photometric data of Cruikshank (1978) and of Slavsky and Smith (1978). Also, we have reconsidered the assumptions underlying the spectroscopic estimates. Photometric periods. The most extensive data are those of Slavsky and Smith, and we have used two methods to find possible periods. The first is the L a f l e r - K i n m a n (1965) method in which a quantity 0 is c o m p u t e d as a function of the a s s u m e d period. 0 is the sum of the squares of the brightness differences b e t w e e n observations of adjacent phase, and we seek a minimum in its o b s e r v e d value by varying the assumed value of the period. This scheme selects periods which produce smooth light curves. The second method is closely related and was d e v e l o p e d by L. Wallace. In this case 0 is c o m p u t e d after the m e a n value o f the brightness observations that fall within an a s s u m e d range of phase (we assumed intervals of 0.1 in phase) has been subtracted. This is analogous to a filtering process and serves to minimize high-frequency variations in the variable as a function of phase for an ass u m e d period. In their published w o r k Slavsky and Smith use the full set of data, which extends o v e r five lunations, to arrive at a period of 18.44 hr. The data set was, however, " c o r r e c t e d " before use in analysis by removing the mean value and trend from

73

the data taken at the separate lunations. This operations was justified by possible changes in the reference stars that were used and b y photometric " w e a t h e r " on N e p t u n e . In the near infrared the latter is k n o w n to show dramatic changes on a monthly time scale (Joyce et al., 1977) and could lead to both amplitude modulation of the photometric signal and also phase shifts b e t w e e n one lunation and the next. The results of such variability, especially when c o m b i n e d with aliasing by the sampling spectrum, could lead to the presence in the data of m a n y possible light curves, each characterized b y similar scatter, but which c o r r e s p o n d to spurious periods. As a result there is, in our opinion, no a priori guarantee that e v e n the best-looking curve will c o r r e s p o n d to the true period. A Fourier analysis of the b r i g h t n e s s - t i m e data would give an equivalent view with m a n y peaks showing up in the p o w e r s p e c t r u m well a b o v e the background noise set by the photometric a c c u r a c y of the data (cf. Fig. 6 in Slavsky and Smith, 1978). In our analysis we have a t t e m p t e d to minimize the a b o v e problems by analyzing individual lunations in Slavsky and Smith's and C r u i k s h a n k ' s data separately and have not applied any ad hoc corrections to the listed data. H o w e v e r , we have also analyzed the full data set in order to provide a direct c o m p a r i s o n to Slavsky and Smith. [We note that the photometric quantity Rp/R. would have been preferred to the quantity R p - R. which Slavsky and Smith used in their analysis; h o w e v e r , they only list the latter quantity. The use of the ratio would, we believe, more effectively assure the r e m o v a l of any changes in instrumental r e s p o n s e that might o c c u r from night to night.] Our results are collected in Table II and in Fig. 2. In the case of C r u i k s h a n k ' s data, we have omitted f r o m consideration the five data points associated with the M a y - J u n e 1977 time period in order to keep the data restricted to a single lunation. We conclude from a c o m p a r i s o n of the various sets of data that there is little rea-

74

BELTON ET AL. T A B L E II POSSIBLE PHOTOMETRIC PERIODS (hr)

Cruikshank (1978) July 1977

Slavsky and Smith (1978) Full data set

March-April 1977

Period

0

Period

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Period

0

19.39 18.15 21.03 14.81 10.34

0.7498 0.7870 0.9573 0.9690 1.1151

18.45 19.16 19.06 10.42 15.25

0.02095 0.02314 0.02343 0.02414 0.02429

16.30 9.67 19.36 18.37 15.56

0.4046 0.4226 0.4639 0.7549 0.9117

June 1977

Period 18.42 12.84 17.11 10.27 13.20

July 1977

August 1977

0

Period

0

Period

0

0.3587 0.4687 0.8189 0.8195 0.8786

21.03 18.85 13.45 8.87 21.82

0.7539 0.8616 0.8738 0.9176 1.0299

18.99 21.03 10.45 20.31 13.01

0.6817 0.7086 0.7152 0.7450 0.8350

" The analysis of the full data set was based on L. Wallace's modification on the Lafler-Kinman technique (see text).

son to accept 18.44 hr as a secure identification of the planetary period. This period is certainly not predominant in the monthly solutions, although periods near to this value do produce reasonable light curves for all except the March-April 1977 Slavsky-Smith data. From the data plotted in Fig. 2 we conclude that any of the listed periods based on the full data set are acceptable. Finally, in comparing the Cruikshank and Slavsky-Smith data for July 1977, we notice that a period of 21.03 hr seems to provide a reasonably low-scatter light curve for both sets of data. A rough comparison of trends in the two sets of data during the few days (July 10-14, 1977) where they were in c o m m o n shows strong similarities in the behavior of the R~, - R. and J - K indexes with time; the significance of this, however, is far from clear. Spectroscopic periods. Underlying the determination of spectroscopic periods reported here is the assumption that the degree of limb darkening exhibited by Neptune is identical to that of Uranus (at least near 4900-5000 ,~ where the measurements are made). If, however, the limb darkening on Neptune is, in fact, less pronounced than has been assumed, then the true period will be longer than the current estimates. For example, if Neptune has a fiat

distribution of light across its disk at 4900 ]k, then the H a y e s - B e l t o n data would imply a period some 6 hr longer ( - 2 1 hr) than the value noted in Section I. Contrary to the position taken by Hayes and Belton in 1977 we note that there now exists a recent and growing body of indirect evidence for significant differences in the atmospheres of the two planets which could, we speculate, be indicators of a substantial difference in their limb-darkening properties. For example, the higher albedo shown by Neptune in the center of strong CH4 bands (Wamsteker, 1973), the different shape of strong methane bands in the region 1-2.5 /~m (Fink and Larson, 1979), and the greater photometric variability of Neptune relative to Uranus (Slavsky and Smith, 1978; Avis et al., 1977; Lockwood, 1978) all could be indicators of a substantial and global presence of aerosols high in N e p t u n e ' s atmosphere that is largely absent on Uranus. On the other hand this aerosol component (should it exist) is nevertheless constrained to be optically thin in order that the similarity of hydrogen quadrupole spectrum between the two planets can be reasonably understood (Trafton, 1974). A coupling of this admittedly speculative evidence for a thin (but, relative to Uranus, substantial) scattering haze high in N e p t u n e ' s atmosphere

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together with the indication in Wamsteker's (1973) work that N e p t u n e ' s geometric albedo is less than that of Uranus in the region 4000-5200 A leads to the possibility that N e p t u n e ' s limb darkening might be substantially reduced from that exhibited by Uranus. We also note that recent measurements of the distribution o f brightness across N e p t u n e ' s disk in the " c o n t i n u u m " near 6800 /~ by Price and Franz (1980) indicate that model brightness distributions that are flat or marginally limb brightened are required to best explain the data. Similar " c o n t i n u u m " observations of Uranus at 6200, 6400, and 7500 ~ , on the other hand, all indicate the presence of some limb darkening (Price and Franz, 1979). While such observations do not definitely prove the speculations made regarding the relative limb darkening on the two planets (e.g., no similar brightness distribution scans exist for Uranus and Neptune at exactly the T A B L E III CORRECTED VALUES OF C,F(~s)" FOR SELECTED LINES ON NEPTUNE (sec ~ × 10 ~) Plate No.

tk~ (deg)

Seeing ~ (arcsec)

1047 1031 1048 1049 1027 1030 1050 1054 1051 1053 1028 1052 1029

20 45 50 70 90 90 110 110 130 130 135 160 180

3.2 1.6 2.7 2.7 2.7 1.1 2.1 3.2 2.9 2.9 2.4 2.1 2.4

Raw data

-0.02 0.39 1.02 0.90 2.64 1.71 2.97 3.05 2.24 2.16 2.18 1.81 1.71

± 0.12 ± 0.41 ± 0.18 ± 0.12 ± 0.36 ± 0.18 _+ 0.30 ± 0.24 -~ 0.24 ± 0.47 ± 0.24 ± 0.30 ± 0.24

Seeing corrected

-0.13 0.79 4.87 4.30 12.60 2.38 8.96 19.58 11.86 11.43 8.35 5.46 6.55

± ± ± ± ± + ± ± ± ± ± ±

0.77 0.83 0.86 0.57 1.72 0.25 0.90 1.54 1.27 2.49 0.92 0.90 0.92

" Refer to H a y e s and Belton (1977) for the precise definition o f this quantity which is proportional to the mean tilt of selected fraunhofer lines on e a c h plate. b The " s e e i n g " refers to the full width at half intensity of G a u s s i a n smoothing function required to explain the distribution of density on the plate across the d isp ersion direction. These new e s t i m a t e s are s y s t e m a t i c a l l y lower by - 0 . 1 arcsec than those listed in H a y e s and Belton (1977).

same wavelengths where the rotation studies were done), they do nevertheless seem to indicate that there is a reasonable chance of their being correct. Thus, we argue that there is a substantial possibility that the spectroscopic periods estimated for Neptune by Munch and Hippelein (1979) and by us are too short by several hours. In conclusion it is our assessment that the period of Neptune is most likely to be longer by several hours than the current spectroscopic estimates o f l l . 2 to 15.4 hr. The exact value is unknown and is not necessarily one of the values indicated by current observations of brightness variation on the planet. Many more occultation chords, which could help tie down the oblateness more accurately, are needed as are much more densely sampled light curves. Imaging o f the planet in regions of CH4 absorption may, however, ultimately yield the most accurate estimate of the period if specific features, now known to exist, can be followed across the disk of the planet for even short (i.e., less than the rotation period) intervals of time (Smith et al., 1979). (b) The P e r i o d o f Uranus

Current estimates of the rotation period of Uranus depend primarily on the spectroscopic technique and an estimate o f the planet's oblateness. Its brightness is apparently not appreciably modulated by rotation ( L o c k w o o d and Thompson, 1978; Avis et al., 1977; cf. Hayes and Belton (1977) for earlier photometric references) and so no modern light curves have yet been published. H o w e v e r , Smith and Slavsky (1979) have announced a photometric period o f 2355T4(_0.2 m) in an abstract. Estimates of the planet's oblateness vary from 0.01 0.01 (Danielson et al., 1972) to the most recent value of 0.033 ___ 0.007 which was obtained from observations of a stellar occultation by the planet (Elliot et al., 1979). Because of this, spread oblateness is not, in our opinion, a predictor of the rotation period that can be used with confidence [as

PLANETARY ROTATION has been a s s u m e d recently by D u n h a m and Elliot (1979) and Franklin et al. (1979)]. The spread of values relative to the formal errors indicate that strong systematic errors m a y affect one or more of the various determinations. The estimate o f oblateness which has the smallest f o r m a l error is that o f Franklin et al. (1979) w h o find a value of 0.021 _ 0.001 b a s e d on a careful reanalysis o f the stratoscope images (Danielson et al., 1972). This value leads to a highly constrained period of 16.8 +__ 0.8 hr which would exclude the period found by H a y e s and Belton (1977) if the possibility of systematic errors is ignored and the formal error in the determination of the oblateness is t a k e n at face value. We, however, prefer to interpret the range of oblateness determinations as indicative of the magnitude of systematic errors that can o c c u r in the various determinations of this quantity and use the oblateness as a discriminator rather than a predictor. We note that the period o f 24 _ 4 hr yields an oblateness in the range 0.011 to 0.016 and that this range of values falls b e t w e e n the results of Elliot et al. on the one hand, and Danielson et al. on the other; this estimate is, therefore, a reasonably probable value for the period if current determinations of oblateness are to be used as a criterion. We conclude that it is not possible at present to choose confidently b e t w e e n the T r a f t o n / H a y e s - B e l t o n period ( - 2 4 hr) on one hand and the T r a u g e r et a l . / M u n c h H i p p e l e i n / B r o w n - G o o d y periods ( ~ 1 6 hr) on the other and that any o f these estimates m a y be close to the actual period. H o w ever, we note that the H a y e s - B e l t o n data on Uranus and N e p t u n e were obtained at the same observing sessions, under identical instrumental conditions, and reduced in an identical fashion; it, therefore, s e e m s reasonable to m a k e the following statements: The period of Uranus is longer than that of N e p t u n e , and that if the H a y e s Belton estimate of N e p t u n e ' s period is proved correct, then so, probably, is their estimate of the period of Uranus.

77

A referee has suggested to us that it would be valuable to include a short discussion of the possibility that the differences b e t w e e n photometric periods and spectroscopic periods might arise as a result of the different spectral regions used in the two types of determinations, the light being reflected from different layers in the atmosphere. Experience with a t m o s p h e r i c motions on Venus, Earth, and Jupiter would indicate that vertical contrasts in zonal flows of global extent are unlikely to exceed 150-200 m sec -~, which for N e p t u n e and Uranus could lead to differences in the apparent photometric and spectroscopic periods of up to 2 hr at the most. A difference of, say, 7 hr, such as exists b e t w e e n the Slavsky and Smith period and the M u n c h - H i p p e l e i n spectroscopic period for N e p t u n e , would require a vertical contrast of zonal motions on a global scale to exceed 1 km sec -~ which, in our opinion, is unlikely to occur. ACKNOWLEDGMENTS We are very grateful to Dr. G. M u n c h and Dr. H. Hippelein and to Dr. F. A. Franklin for c o m m u n i c a t ing their results to us prior to publication. REFERENCES A v i s , C. A., SMITH, H. J., BERGSTRAHL,J. R., AND SANDMAN, W. H. (1977). Photometric determination of the rotation period of U r a n u s . Bull. A.A.S. 9, 472-473. BROWN, R. A., AND GOODY, R. M. (1977). The rotation of U r a n u s . Astrophys. J. 217, 680-687. BROWN, R. A., AND GOODY, R. M. (1979). The rotation of U r a n u s . II. Astrophys. Ji CRUIKSHANK,D. P. (1978). O n the rotation period o f Neptune. Astrophys. J. Lett. 220, L57-160. DANIELSON, g . E . , TOMASKO, M. G., AND SAVAGE, B. D. (1972). High resolution imagery of U r a n u s obtained by stratoscope II. Astrophys. J. 178, 887900. DUNHAM, E., AND ELLIOT, J. W. (1979). T h e rotation period of Uranus. Bull. A.A.S. 11, 568. ELLIOT, J. L., DUNHAM, E., MINK, D. J., AND CHURMS, J. (1979). T h e radius and ellipticity o f U r a n u s from its occultation of S A O 158687. Submitted. FINK, U . , AND LARSON, n . P. (1979). T h e infrared spectra of U r a n u s , N e p t u n e , and Titan from 0.8 to 2.5 Microns. Astrophys. J. 223, 1021-1040.

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