INFORMATION LIMIT OPTIMISATION TECHNIQUES

APPLIED TO RECENT PHOTOMETRY OF PLUTO

TIMOTHY BANKS

Physics Department, Victoria University, Wellington, New Zealand

and

EDWIN BUDDING

Carter Observatory, Wellington, New Zealand

(Received 14 February, 1990)

Abstract: An algebraic approach originally intended for the heuristic modelling of the maculation effects of chromospherically active stars has been used to model the rotational modulation of Pluto’s light. Two dark spots (30.0” r 0.8” and 16.4” ? 1.4” in radius, at longitudes 3.4 F 3.2 and 138.0 ?z 5.2 degrees, and respective latitudes -35” and -16”) were found to model the 1982.2 data of Binzel and Mulholland (1983) best. These parameters are in reasonable agreement with the other published models of Pluto’s surface for the data, however this model can not be well reconciled with older data sets. A possible solution is dynamic behaviour of the dark features themselves, increasing in radius as perihelion is approached. We stress a minimum in the introduction of ad hoc surface feature hypothesis in this treatment.

Introduction

It has been known for a number of years now (Walker and Hardie, 1955) that the planet Pluto exhibits photometric variations, chiefly on an approximately 6.4 day timescale. The possibility of modelling this variation with computers using suitably defined areas of different reflective properties on a spherical body has recently arisen (Marciallis, 1983,1984,1988). In this article we propose to follow a generally similar approach to that of Marciallis (1983) but using our recently described Information Limit Optimization Technique (ILOT) (Banks and Budding, 1990), which was originally developed for chromospherically active stars (Budding, 1977).

The periodic variation of Pluto’s light curve, which has been attributed to surface features rotating through our line of sight, was first noted by Walker and Hardie (1955). Ten years later Hardie (1965) discovered a secular variation in the light curve’s amplitude and mean magnitude, which was confirmed by later workers (Tedesco and Tholen, 1980; Neff et al., 1974; Kiladze, 1967). The actual asymmet- ric (see Figure 1) shape of the light curve remained essentially constant throughout the literature, suggesting that static surface features were observed, rather than a cloud-like feature. In this paper we compare our results with those of Marciallis (1983) who used a similar, but finite element, spot approach to Pluto’s photometric variations.

Earth, Moon, and Planets 49: 15-23, 1990. 0 1990 Kluwer Academic Publishers. Printed in the Netherlands.

The Data and its Model Fitting

A x2 minimisation program using the algebraic theoretical maculation wave fitting function of Budding (1977) was used to model the standard Johnson B light curve obtained by Binzel and Mulholland (1983) during the interval January to June 1982. This particular data set was chosen because it is the most recent and accurate in the literature, and the minima were deep, allowing better determination of the spot parameters. The magnitudes have been normalised to the mean opposition distance of 39.5 AU, with the phases calculated using Hardie’s (1965) period of 6.387d and Tedesco and Tholen’s (1980) epoch of JD 24444240.59. Two different depth minima were obvious in the light curve, which were taken to be the effect of two reasonably large well separated spots of rather different radii.

We assumed that the photometric variations are due to dark spots rotating through our line of sight, instead of light ones against a dark background. Relative, rather than absolute, albedos were used so as to remove a dependence on Pluto’s actual radius, which is still rather uncertain at the time of writing (Beatty, 1987). While the surrounding regions’ albedos were set to unity, those of the two spots were set equal to zero thus producing minimum area spots.

A plutocentric coordinate system was used, with North defined as the direction of Pluto’s rotational angular momentum vector. This is not the same as the IAU’s definition (Davies, 1980), but it is simpler in our context. Unfortunately, our procedure cannot automatically differentiate which hemisphere contains the spots for inclinations near 90”. The instantaneous value of the inclination was reached using the distance and phase angle information of the “Astronomical Almanac”. These indicate that since Pluto’s discovery in 1930, until very recently, the southern hemisphere has been tilted towards us, although this angle has decreased, bringing the planet ‘side on’ (axis perpendicular to the light of sight) to us in 1987/88. This, combined with the increasing amplitude of the light curve, suggests that it is the southern hemisphere which contains the spots.

Banks (1989) has shown that using light spots in the place of dark ones, while lowering the reference brightness, simply shifts the centres of the spots by 180”, and scales their areas by an appropriate factor depending on the selected albedo ratio (see also Banks and Sullivan, 1990). To explain the observed secular dimming of Pluto, light regions would have needed to be rotated out of sight as the inclination increased, however this would have also implied that the light curve’s amplitude should decrease. This is the opposite of what has been observed, explaining why dark spots were preferred.

The ILOT procedure has the following points to commend it: (1) The relatively simple and compact algebraic form of the fitting function

allows large regions of 2 parameter space to be examined at a low computing cost.

(2) The 2 Hessian (see, e.g., Bevington, 1969) can be relatively simply evalu- ated in the vicinity of the adopted minimum. Inspection of this matrix, and in

INFORMATION LIMIT OPTIMISATION TECHNIQUES 17

particular its eigenvalues and eigenvectors, permits a valuable insight into par- ameter determinacy and interdependence.

(3) The Hessian can be inverted to yield the error matrix (cf. e.g., Bevington, 1969). This must be positive definite if a determinate and ‘unique’ optimal par- ameter set is to be evaluated. Strict application of this provision prevents ‘over- parameterisation’ of the data (see also the appendices to Budding and Najim, 1980, or Banks and Budding, 1990).

The program uses three different standard optimisation techniques-parabolic interpolation in ‘step by step’ (single parameter) and conjugate direction (‘Powell’) modes, and ‘vector’ (many parameter) mode in searching for improvements in the x2 value. The sequence switches between these options depending on the convergence rate and certain user-set control quantities (for specifics, see Banks (1989); and Adby and Dempster (1974) for a more general background).

The limb-darkening value was arbitrarily set to zero, despite there being several assumptions of it being extreme in the literature (see Marciallis, 1988). We justify our choice by noting that the limb darkening effect is normally interpreted (for stars) in the context of heat transfer through a stratified atmosphere (see, e.g., Batten, 1973). We believe that Pluto’s atmosphere is not dense enough for it to cause an appreciable variation, and feel at liberty to assume a generally evenly deposited isotropic (‘Lambert’s Law’) surface scatterer for the bright and major component of the surface (‘frost’).

A further assumption was that the origin of the variations is Pluto alone. This is supported by Marciallis et al. (1987) who commented that while 20% of the light from ‘Pluto’ is due to Charon, the wave is essentially due to Pluto. They observed that this variation persisted even when Charon was completely eclipsed.

Results

An initial fit was attempted using an ‘ad hoc’ inclination of 90” optimising the latitudes, longitudes, and radii of both the spots. The indeterminacy that resulted was removed only when both the spot latitudes were removed from the variable

TABLE I

Fixed middle latitude soots solution

Al = 138.4 * 5.7” A2 = 3.3 k 3.3” -yl = 19.4 k 1.9” 7-J = 34.9 + 1.0” p1= -45” ,& = -4.5” AI = 0.025 ,yz = 23.1

The symbols used in this paper’s tables are those introduced by Budding (1977) and Budding and Zeilik (1987). A is longitude, platitude, y radius (all in degrees), and the subscript identifies the spot. AZ is the adopted percentage observational error.

18 TIMOTHY BANKS AND EDWIN BUDDING

TABLE II

Grid search optimum parameters

AI = 138.0 f 5.2” A2 = 3.4 -c 3.2” yl = 16.4 + 1.4” y2 = 30.0 + 0.8” PI = - 16.3” p* = -34.4” Al = 0.025 2 = 21.8

parameter set. They were locked to middle latitude values, which has been a standard procedure in modelling starspots (Budding and Zeilik 1987) in similar, rather indeterminate, situations. The parameters of this fit are given in Table I.

This fit demonstrated that we could determinately model the rather sparse data, and so it was decided to use a grid search, with the correct inclination of 79”, in which optimised solutions would be attempted over a range of fixed spot latitudes (0 to 1.4 radians in 0.1 steps), looking for a determinate global minimum. The best fit parameters of this search are given in Table II.

Discussion

The results presented in the foregoing section are not too distant from those of Marciallis (1984), who found that two dark spots of radius 48” and 28”, both

1.45

1.40

l-35

l-30

c

g 1.25

F z

l-20

l-15

1.10

1.05 too

I I 7 I -50 0 50 100 150 200 250 300 350 400

PHASE (DEGREES)

Fig. 1 The model light curve (smooth line) produced by the optimum grid search parameters for the 1982 light curve is plotted against the data (circles).

INFORMATION LIMIT OPTIMISATION TECHNIQUES 19

located at latitude -23, and separated by 134” of longitude, satisfied the data. Our study found that spots at this latitude, assuming an inclination of 79”, gave a solution y of 22.7 (g/u= 1.135), which is not much greater than our optimum fit. His spot latitudes are halfway between those found by our search; and the apparent spot coverage, once corrected for the albedo difference (Marciallis set the spots’ relative albedo to OS), is in good agreement between the two models. We would like to stress that our analysis is independent, and was not influenced at all by the results of Marciallis, e.g. they were not used as starting parameters.

Marciallis found that his solution generally explained the increasing amplitude of the distortion wave with time that has been observed. In a check to see if such variations could be more than a function of Pluto’s inclination (as Marciallis had proposed), the parameters derived for the 1982.2 data were used to model three older data sets, obtained in 1975 (Lane, Neff, Andersson, and Fix, 1955), 1964 (Hardie, 1965), and 1953-55 (Walker and Hardie, 1955). Initially only the mean ‘unspotted’ light level was optimised, with the derived 1982 parameters and an appropriate inclination for the time of observation. The 2 values for these fits are given in Table III. It was found that despite the changing inclination the model did not account very well for the older data (see Figure 2). This can be seen particularly for the 1975 and 1964 data, where the reduced 2 (g divided by the degrees of freedom V) increases from 1.04 for 1982 to 3.26 and 1.48, respectively. The sparse reduced data of 1954 suggest that the adopted level of observational error (2.5%) is too high, at the 85% confidence level (Bevington, 1969). It is difficult to be definitive about what the real observational errors are with these older data sources. What we can show clearly however, in Table III, is the great relative improvement (r, being x$/x:) when we allow the sizes of the spots to relax, i.e. we attempt to seek for any spot “evolution” over this thirty years time interval. We do find a trend towards smaller spot size with increasing solar distance in these determinate fits, suggesting that the features may be a time-dependent

TABLE III

Fits to the older data sets

Year n * * Xl X2 Yl Y2 r

1982 26 21.8 21.8 16.4 2 1.4” 30.0 + 0.8” 1.00 1975 12 29.5 22.7 9.6 ” 5.1” 25.9 + 1.9” 0.77 1964 14 16.3 7.0 15.5 2 2.2” 22.8 ” 1.5” 0.43 19.54 21 12.3 6.4 14.9 + 2.3” 19.3 t 1.6 0.53

The observational error was assumed to be 2.5% of the total intensity for all the data sets. xl is the goodness of fit corresponding to the regressed 1982 model, while ,$z corresponds to fits where both spot radii (which are also given in the table) were included as free parameters. n is the number of data points in each light curve. There is a slight ‘hiccup’ in the trend for the smaller spot in 1975, although it possesses a large error. Banks (1989) noted that slight variations in a larger spot can have quite dramatic effects on a smaller spot’s parameters.

TIMOTHY BANKS AN,, EDWIN BUDDING

Fig. 2 Despite appropriately regressing Pluto’s inclination, the fixed spot model derived from the 1982 data, does not ‘fit’ well the older data (in this example the 1964 set). If the spot radii are allowed

to vary a closer fit was reached.

phenomenon varying in relation to this distance. This pattern appears more clear for the larger, and more easily resolved, spot.

This by itself does not explain the secular dimming of Pluto over the same period. Such a ‘DC’ contribution, as opposed to the ‘AC’ maculation effect, is usually attributed (see, e.g., Banks and Budding, 1990) to an essentially uniform banding, as the effect is independent of longitude. If we assume a basically static surface model, then the dimming must be a function of the inclination (see Figure 3). Thus either a dark northern circumpolar region has rotated into view since observations began, or a light southern cap has moved out. Marciallis (1988) opted for the later view, but with a highly reflective cap at each pole, presumably consisting of ‘fresh’ methane ice. However we would like to note that a global albedo decrease, connected with methane release near perihelion, could also be responsible for the secular dimming. A scenario is that this gas release, thickening the atmosphere, results in the exposure of a darker layer underlying a thin methane crust, either as a thinning of the layer or a complete exposure of isolated regions. The larger scale exposure of the darker sublayer is then the cause of the maculation effect, with the lesser, more general, depletion being responsible for the secular variation. Such an explanation removes the need to resort to further assumptions, such as polar caps, that are beyond the present information content of the data. A simple-minded association of the bright ‘frost’ layer with precipitated methane, which might have some heuristic appeal, appears on more detailed investigation,

INFVKMATION LlMlT “PTIMlSATlVN TECHNIQUES

22 TIMOTHY BANKS AND EDWIN BUDDING

however, not to infer easily the likely physical processes involved (see, e.g., Stern et al., 1988),

It was not possible to determinately model polar caps of the kind considered by some of the authors refered to. We have already seen that uniqueness problems were encountered when even one spot latitude was included as a free parameter. A polar cap requires another parameter (radius), producing exactly the same problem, as well as requiring an additional assumption about its albedo.

Conclusions

(1) The finite element dark spot model of Marciallis (1983) is generally sup- ported for the 1982 data by our algebraic approach, which is somewhat more versatile in that parameter errors are calculated and both the spot latitudes can be varied independently. We can not confirm his polar caps using this model, which appears to exceed the information content of the data. Bearing in mind our comments above about light and dark spots, Buie and Tholen’s (1989) 22 free- parameter model is essentially equivalent in its photometric consequences to Mar- ciallis’ (see also p. 29, Buie and Tholen, 1989), and so to our own.

(2) The eccentricity of Pluto’s orbit results in a dramatic variation in the amount of insolation experienced between aphelion and perihelion. We postulate that this results in the sublimation of methane ice near perihelion, forming both the surprisingly large atmosphere (see Millis et al., 1988) and the increased maculation effect. Our ILOT solution for the 1982 data plus a changing inclination is not adequate for the older data sets. We reconcile this discrepancy as an increase in the spot radii as perihelion is approached, as might be expected in such a model (c.f. also Elliot et al., 1989).

(3) A major point of this paper is that it is important not to over-parameterise the model, which will inevitably result in a ‘better’, but rather imprudent, fit to the data. We also note that it would be highly useful to combine the sort of analysis provided in the foregoing with analysis of the Pluto/Charon mutual event photometry, which will provide further constraints and information.

Acknowledgements

The authors would like to acknowledge the fruitful discussions and suggestions of Messrs Frank Andrews (who also kindly presented this paper for the authors at the 1990 RASNZ Conference) and Graham Blow, both of Carter Observatory.

References

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Astronomical Almanac for the Year 1988, The United States Government Printing Office, Washington D.C.

INFORMATION LIMIT OPTIMISATION .TECHNIQUES 23

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