THE ROTATION PERIOD AND LIGHT-CURVE AMPLITUDE OF KUIPER BELT DWARF PLANET 136472 MAKEMAKE (2005 FY9)

The Kuiper Belt is interesting in several ways. It is a relatively unexplored region of our solar system. It may have had profound effects on the rest of the solar system, from influencing the migrations of the giant planets, to delivering impactors to Earth and the rest of the inner solar system (Gomes et al. 2005). Its chemical and dynamical properties convey information about the formation and early evolution of the solar system. Finally, it is an analog to cold debris disks that are now being discovered by the Spitzer Space Telescope around other stars (Bryden et al. 2006).

Since even the largest Kuiper Belt objects (KBOs) can only be marginally resolved even by the Hubble Space Telescope (HST), photometric monitoring is an important tool for their study. For all objects, it can reveal their shape and rotation period (Sheppard & Jewitt 2002; Ortiz et al. 2006). For larger objects such as Pluto, it can indicate the distribution and color of surface albedo features. Over a period of decades, it can reveal the change in sub-Earth latitude, and possibly evidence of seasonal change in albedo features (Buratti et al. 2003). Information about shape and rotation constrains the impact history and physical properties (i.e., density and material strength) of the KBOs (Trilling & Bernstein 2006; Rabinowitz et al. 2006), which in turn provides tests of theories of solar system formation and early evolution. Information about color and albedo features, especially when combined with spectra, provides clues about the surface and atmospheric chemistry of the larger objects (Buratti et al. 2003; Brown et al. 2005; Licandro et al. 2006b).

This is an era of remarkable discoveries in the Kuiper Belt, with the three brightest known objects other than Pluto (136472 Makemake, 136108 Haumea, and 136199 Eris) all having been discovered within the last six years. Of these three, only Haumea has a confident published rotation period at present, and its rotational light curve, indicating a centrifugally elongated, highly ellipsoidal 1000 km object rotating once every 3.9 hr, makes it one of the most fascinating and bizarre objects in the solar system (Rabinowitz et al. 2006).

We present the first confident determination of the rotation period and rotational light-curve amplitude of 136472 Makemake. In Section 2, we describe our observations, in Section 3 we detail our photometric reduction strategy, and in Section 4 we present our period determination methods and discuss the resulting periods. We discuss the implications of our results in Section 5, and we present our conclusions in Section 6.

In 2006 February, we observed 136472 Makemake (then called 2005 FY9) in the U, V, and I bands for four nights through bright moonlight using the 1.54 m Kuiper Telescope. We were unable to determine the rotation period from these data, but the upper limits that we could set on the amplitude of the object's photometric variability suggested that the project was very difficult. Accordingly we applied for and received eight nights of lunar dark time in the spring of 2007. We experienced remarkably good weather, acquiring useful data on five nights in February and two in April. The 2007 data are so far superior to those from 2006 that we have based our light-curve analysis only on our 2007 images.

We observed Makemake on the UT dates 2007 February 19, 21–23, and 26, and 2007 April 23–24, using the University of Arizona's 1.54 m Kuiper Telescope on Mt. Bigelow, just north of Tucson, AZ. Our instrument was the Mont4K CCD, a 4096 × 4096 pixel imager yielding a 9.7 arcmin square field of view. We used the CCD in 2 × 2 binning mode, obtaining a pixel scale of about 0.28 arcsec pixel⁻¹. Our 2007 observations are summarized in Table 1.

Because our 2006 observations had shown the light-curve amplitude to be very small, we planned our 2007 observations to obtain the largest possible amount of consistent, high-quality photometry. We spent a bare minimum of time on calibration stars during the time Makemake was observable each night, and used the V band almost exclusively to get a very large data set with consistent photometry. We used an integration time of 500 s, which yielded bright but unsaturated images of Makemake and appropriate comparison stars in the same field.

We processed our data using dark subtraction, flat fielding, correction of bad pixel regions, removal of cosmic rays using two iterations of our implementation of the Laplacian edge detection algorithm described in the excellent work of van Dokkum (2001), and finally removal of isolated deviant ("hot") pixels (e.g., residuals from cosmic rays).

We paid special attention to the construction of our flat frames, because these were critical to the success of our photometric strategy. In the course of our observations, we obtained both dome and twilight sky flats. Dome flats can be taken in large numbers and averaged to obtain extremely low noise. Conditions appropriate for taking sky flats occur only briefly, so averaged sky flats are always noisier. Dome flats, however, do not provide as reliable a measurement of the true illumination of the detector in real observing conditions. We processed our data using three different flat-fielding strategies: pure dome flats, pure sky flats, and composite flats. These last were constructed by first creating a sky/dome ratio image, then smoothing it, and then multiplying the original dome flat by the smoothed ratio, to produce a final image with the low noise of a dome flat but the correct illumination of a sky flat at low spatial frequencies. As expected, the dome flats produced substantially more scattered photometry than the other two strategies. The sky and composite flats produced very similar results, but the sky flats were slightly superior. We have therefore used images processed with sky flats throughout our analysis. Any flat-field gradients or other defects in the sky flats are observed to be far below what would be required to explain the observed variability of Makemake.

On the night of UT February 26, Makemake passed very close to two faint stars, which caused a spurious brightening of a few percent in our raw photometry. To solve this problem, we first stacked a large set of images on which Makemake was not close to the stars, obtaining a very low-noise image on which the faint stars could be accurately measured. Using this image, we found precise values for the positions and brightnesses of the faint stars relative to a nearby, much brighter star. We then processed every February 26 image by subtracting an appropriately scaled and shifted version of the point-spread function (PSF) of the brighter star on that image from the exact location of each of the faint stars we wished to remove. We tested the effectiveness of the subtraction by creating a new stacked image of the altered data. The stars had essentially vanished, as did the deviant signature in our Makemake photometry, leaving the February 26 data entirely consistent with the rest of our nights.

Figure 1 shows all the data from our observations on 2007 February 19, 21, 22, and 23 stacked to make a single image, with the path of Makemake appearing as four approximately collinear streaks moving toward the upper right.

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3.1. Acquiring the Relative Photometry

Our focus in this project was extremely accurate relative photometry. For each night, we identified a set of reference stars clearly visible on all images from that night. This set of reference stars was determined individually for each night, but because Makemake moves very slowly there was considerable overlap from one night to the next.

We obtained aperture photometry of all the reference stars and the science target. We tested the stars for variability by ratioing each to the sum of all the others, and calculating the normalized rms scatter of the result. Since fainter stars ought to have a larger normalized scatter due to statistical photon noise and background noise, this normalized rms should decrease monotonically as the brightness of the stars increases. Stars that deviated from this expected monotonic decrease were flagged as possible variables and, if the deviation was significant, removed from the reference star set. For most nights, extending this analysis to include Makemake itself resulted in its also being flagged as a variable "star," as we would expect.

Having removed clear variables from our reference star set, we obtained relative photometry in which the brightness of Makemake was ratioed to the summed brightness of all the reference stars. There was a considerable overlap in reference star sets from one night to the next. This allowed us to make all the relative photometry in February, and all that in April, internally consistent with no guesswork. Doing this as accurately as possible was very important. Our first concern was to find the ratio of the summed flux over all the reference stars used in one night to the summed flux over all those used in the next night. As an example, we will take nights 1 and 2. Let i index images and j index stars in a given night, where F_ij will be the observed flux from star j on image i. Let s₁ represent the set of all reference stars for night 1, s₂ represent the set of all reference stars for night 2, and s₁₂ represent the subset of stars that was used for both nights 1 and 2. For each image i in night 1, we can then form the ratio

$$\begin{equation} R_{1i} = \frac{\sum _{j \in s_{1}} F_{ij}}{\sum _{j \in s_{12}} F_{ij}}. \end{equation} \tag{ 1 }$$

Of course, this simply ratios the sum over all reference stars used on night 1 to the sum only over those shared between nights 1 and 2. Now let R₁ be the average of R_1i over all the images in night 1:

$$\begin{equation} \mathbf {R_1} = \overline{R_{1i}}. \end{equation} \tag{ 2 }$$

This R₁ will then be an extremely accurate estimate of the true ratio of the summed flux of the two star sets s₁ and s₁₂. We can proceed similarly for night 2, letting i now index images in night 2 rather than night 1:

$$\begin{equation} R_{2i} = \frac{\sum _{j \in s_{2}} F_{ij}}{\sum _{j \in s_{12}} F_{ij}}. \end{equation} \tag{ 3 }$$

And, again, R₂ will be an average over all the images in night 2:

$$\begin{equation} \mathbf {R_2} = \overline{R_{2i}}. \end{equation} \tag{ 4 }$$

Now we define a new ratio

$$\begin{equation} \mathbf {C_2} = \frac{\mathbf {R_2}}{\mathbf {R_1}}. \end{equation} \tag{ 5 }$$

C₂ is of course the ratio of the summed flux over all night 2 reference stars (the set s₂) to the summed flux over all night 1 reference stars (the set s₁). It has been constructed exclusively using the ratios of stellar fluxes measured on the same images. The effects of seeing, air mass, and atmospheric transparency variations cancel out, yielding a remarkably precise value.

We have shown how we can precisely measure the ratio of the summed flux of all reference stars used on night 2 to that of all reference stars used on night 1, assuming the sets overlap somewhat. In general, there was extensive overlap, well over 50%. Of course, there might be little or no overlap between, say, the first night and the fifth night. Rather than trying to establish the ratio of reference star sums across such a large gap directly, we calculated them exclusively from one night to the next, so that, e.g., C₃ was the ratio of the sum over stars used on night 3 to those used on night 2, C₄ was the ratio of stars from night 4 to stars from night 3, etc. Finally, the relative photometry for the target from, say, image i on night k was calculated by

$$\begin{equation} T_i = \left(\frac{F_{{\rm targ},i}}{\sum _{j \in s_k} F_{ij}} \right) \cdot \prod _{n=2}^{n=k} \mathbf {C_n}. \end{equation} \tag{ 6 }$$

Here, of course, i indexes images in night k, j indexes stars in the reference star set for night k, and n indexes nights. F_targ,i is the flux from Makemake on image i of night k, and T_i is the final relative photometry result for Makemake on image i. This procedure gave us a consistent scaling for all the photometry from February, and all that in April, assuming the reference stars were nonvariable (as we had already tested), but making no assumptions about the brightness of Makemake itself. The longer gap between the February and April data was bridged using images of the April field taken during our February run, under excellent weather conditions when both fields were high in the sky. The February–April bridge was the only step where we had to use a ratio of two sets of stars that were not both measured on the same images. A larger aperture was used for this final ratio, to minimize the effects of seeing variations between the different images. Our final step was simply to normalize the relative photometry.

3.2. Selecting the Final Aperture and Reference Stars

We used a photometric aperture of 13.0 pixel (3.6 arcsec) radius for our first-pass relative photometry of Makemake. We constructed a Lomb–Scargle periodogram of this preliminary data set using the computer program subroutine period.c from Press et al. (1992). This routine identified clear sinusoidal signals at about 5.9, 7.8, and 11.4 hr periods. These values are consistent with a true 7.8 hr period flanked by positive and negative diurnal aliases. We made folded light curves at each period, and found that the 7.8 hr period also showed the lowest scatter. We adopted this period as our preliminary model. We tried different photometric apertures, and found that an aperture of 10.5 pixel (2.9 arcsec) radius minimized the scatter from the 7.8 hr sinusoidal fit. We adopted this as our aperture for our final photometric analysis.

We experimented with different subsets of our initial selection of nonvariable reference stars. One concern was that reference stars of different colors might introduce systematics into our relative photometry, due to the color-dependent extinction of Earth's atmosphere. We had acquired a few B-band images each night, and so were able to obtain the B − V colors of our reference stars. Our investigation showed no evidence of measurable color-dependent atmospheric extinction, so we can be confident that this is not a source of systematic error in our photometry. From 10 semirandom trials with different subsets of reference stars, we chose the subset that produced the lowest scatter from the 7.8 hr sinusoidal fit for our final photometric analysis. The majority of bright (V < 17.0) stars present in the initial nonvariable sample were included in this final, optimized set.

For several reasons, our choice of a 7.8 hr period sinusoid as our standard model for determining the optimal reference star set is unlikely to have meaningfully biased the final data in favor of this period. First, we had already screened the reference stars for variability before the initial determination of the period, and the final set of reference stars we adopt includes the majority of bright stars identified as nonvariable at first. Also, a 7.8 hr period with a sinusoidal variation appears clearly for every reasonable choice of aperture and reference star set, and it appears in both the February and April data when these are analyzed separately. Our tuning of the photometric aperture and the reference star set simply reduced the random scatter in the final photometry.

Table 2 and Figures 2 and 3 present our final photometry, as optimized by the process above. The figures present photometry after correction for changes in the Sun–object–Earth phase angle, the geocentric and heliocentric distances, and the light-travel time; the table presents uncorrected values. We have used distance information from the Minor Planet and Comet Ephemeris Service (http://www.cfa.harvard.edu/iau/MPEph/MPEph.html) in applying these corrections. Figure 2 shows our 2007 February photometry data, with sinusoidal fits at the 5.9, 7.77, and 11.4 hr periods. Figure 3 presents the data and fits for 2007 April. The figures support our initial impression that 7.77 hr is the true period, while the others are diurnal aliases in which either one too few or one too many cycles occur during the diurnal gaps in our observations.

Table 2. Relative Photometry

Time (hr)	Relative Flux	Heliocentric Distance (AU)	Geocentric Distance (AU)	Phase (deg)
4.8969	0.9947	51.983822	51.188069	0.6514
5.0485	0.9928	51.983823	51.188032	0.6513
5.2000	0.9864	51.983824	51.187995	0.6513
5.3617	1.0069	51.983826	51.187955	0.6512
5.5130	1.0017	51.983827	51.187918	0.6512
5.6641	1.0021	51.983828	51.187881	0.6511
5.8275	1.0176	51.983829	51.187841	0.6510

Notes. Time is measured from 00:00 UT 2007 February 19 and is not corrected for light-travel time. Flux is not corrected for distance or phase.

Only a portion of this table is shown here to demonstrate its form and content. Machine-readable and Virtual Observatory (VO) versions of the full table are available.

Download table as:  Machine-readable (MRT)Virtual Observatory (VOT)Typeset image

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3.3. Phase Slope

Our measurements were timed near opposition to get the longest possible observing time per night and optimize our ability to constrain the period. For this reason, they do not span a large range in Sun–object–Earth phase angle: the range is only 059–084. However, our photometry is sufficiently accurate to derive a meaningful phase slope: 0.037 ±  0.013 mag/°. This agrees with the result of Rabinowitz et al. (2007), who found a slope of 0.054 ±  0.019 mag/°. These measurements fit well into the overall pattern of shallow phase slopes for large KBOs and steeper slopes for small ones (Rabinowitz et al. 2007). For comparison, the V-band phase slope of Pluto is 0.032 ± 0.001 mag/° (Buratti et al. 2003), that of Eris has been measured at 0.105 ±  0.020 mag/° (Rabinowitz et al. 2007) and 0.09 ± 0.03 mag/° (Sheppard 2007), and that of Haumea is 0.110 ± 0.014 mag/° (Rabinowitz et al. 2007). The phase slopes of smaller KBOs are typically considerably steeper, ranging up to about 0.25 mag/° (Rabinowitz et al. 2007). Figure 4 shows our data, after correction for changing geocentric distance and the subtraction of a smoothed model of the rotational light curve, plotted against Sun–object–Earth phase angle.

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3.4. Tests for Long-Term Variability

To make our final analysis of periodic variations in the photometry of Makemake, we used photometry from the optimized aperture and reference star set discussed in Section 3.2. These data are also plotted in Figures 2 and 3. This data set was corrected for Makemake's changing Sun–object–Earth phase angle and heliocentric and geocentric distances. We also applied a light-travel time adjustment to our photometric timing.

The second line from the top in Figure 5 shows the Lomb–Scargle periodogram for our final data set. Clear peaks with vanishingly small false alarm probability are seen at a period of 7.7716 hr and at diurnal aliases of 5.8934 and 11.4072 hr, with the latter alias being much the stronger of the two. There is no evidence for the 11.24 hr periodicity suggested by Ortiz et al. (2007). To test if the strong 11.4 hr alias might actually be the true period, we created synthetic data sets with the same time sampling as our real data, but with the actual photometry replaced with values from sinusoidal fits to our data at the 7.7716 hr and 11.4072 hr periods, plus pure Gaussian noise with σ = 0.9%. The top line and the line third from the top in Figure 5 are the periodograms from these two synthetic data sets. The relative strengths of the different peaks in the real data match excellently with those from the synthetic data with a 7.77 hr period, and do not match those from the 11.4 hr synthetic data set.

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We also analyzed our data set using the phase dispersion minimization (PDM) method (Stellingwerf 1978), which, unlike the Lomb–Scargle periodogram, does not assume that the data will be approximately sinusoidal. The PDM method scans a broad, finely sampled range of periods, and identifies the one producing the minimum dispersion, parameterized by the θ statistic, which is a measure of the deviation of the actual data from a smoothed light curve folded to the given period. To make this smoothed light curve, one first folds the data and then smooths it using a sliding boxcar average over a fixed fraction of the period. The length of our boxcar was 0.2 periods: this was the shortest value that gave clean, non-jagged curves.

The results of applying the PDM method to our data and to the two synthetic data sets previously discussed are shown in the top three curves of Figure 6. Again, the 7.77 hr period is strongly favored, and the 11.4 hr synthetic data do not reproduce the observed relative amplitudes of the different peaks. The real data show a strong peak at 15.54 hr, which is twice the dominant period and corresponds to a double-peaked light curve. The highest peak near 11 hr has shifted to 11.6878 hr, rather than 11.4072 as in the periodogram. This 11.69 hr period was not seen in the periodogram because the resulting light curve is strongly non-sinusoidal. The fact that the same peak appears in the synthetic 7.77 hr data suggests it is an artifact of the sampling. The highest peaks near 7.77 and 11.4 hr in the PDM diagram are not significantly shifted from the same peaks in the periodogram analysis.

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Our dominant peak near 7.77 hr is divided into many subpeaks separated by about 0.04 hr, which correspond to different integer numbers of periods elapsing between our February and April observing runs. This is shown in Figure 7. The peak at 7.73 hr is very close in amplitude to the dominant peak in the periodogram, but the PDM analysis is decisive in preferring the 7.77 hr peak.

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Even with our careful night-to-night calibration, flat-field errors and other effects could potentially impose slight night-to-night shifts in our photometry. Surprisingly, shifts of this type can move power from one peak to another in a periodogram or PDM analysis (J. L. Ortiz 2009, private communication). Accordingly, we calculated the average nightly offsets from our best-fit smoothed model, and found an rms average offset of 0.442%. We performed a Monte Carlo simulation in which random offsets following a Gaussian distribution with σ = 0.442% were added to each night of our real data. The Monte Carlo simulation involved 1000 realizations, each of which was analyzed with both the periodogram and the PDM techniques.

The periodogram analysis preferred a period near 7.77 hr in 76.5% of cases, a period near 7.73 hr in 18.4% of cases, and a period near 11.4 hr in 4.8% of cases. It gave bizarre and unreasonable results in 0.3% of cases. The PDM analysis gave a period near 7.77 hr in 91.6% of cases, a period near 7.73 hr in 0.2% of cases, a period near 11.4 hr in 6.3% of cases, and other periods in 1.9% of cases. The lowest line in Figures 5 and 6 is from one of the rare realizations in this Monte Carlo simulation where the 11.4 hr period was preferred. Note that even in this case, the peaks do not match well with the 11.4 hr synthetic data: in particular, the 5.9 hr alias is too strong.

In only 4.8% of cases did both PDM and periodogram methods yield a period near 11.4 hr. Since the 7.77 hr period is preferred by both methods in the real data set, it appears we can rule out the 11.41 hr period at roughly the 95.2% confidence level based on this experiment. The PDM method should be sensitive to any real periods in the data. Since it finds a 7.73 hr period in only 0.2% of cases, it also seems unlikely that this can be the real period.

Ortiz et al. (2007) carried out a program of intensive photometry on Makemake similar to our own. They preferred a period of 11.24 hr, but peaks near 11.41 and 7.8 hr do appear in their Figure 1, with the 11.41 hr peak specifically mentioned in the caption. Thus, there is an indication of the dominant period we detect in their data as well.

To complete our analysis, we present in Table 5 a list of all the periods that have been considered for Makemake: the 7.77 hr dominant period, the 7.73 and 11.41 hr periods we have already largely dismissed, the 11.24 hr period preferred by Ortiz et al. (2007), the 11.69 hr period identified in our PDM analysis, and the 15.542 hr double-peaked period from our PDM analysis.

In Figures 8–12, we compare the folded light curve at our dominant period of 7.77 hr against light curves constructed from our data folded at different periods. In every case, the light curve is fitted by a smoothed model such as is used in the PDM analysis, and the rms scatter from this model is given. A total of 1.5 periods are shown to allow examination of any possible discontinuity.

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Figure 8 compares our dominant light curve with the strong alias at 11.41 hr. We have already argued based on the Monte Carlo simulation that this period is ruled out at the 95% confidence level; visually, as well as mathematically, the fit is decidedly poorer.

Figure 9 compares the dominant period with the subpeak at 7.73 hr. In the Monte Carlo simulation, the 7.73 hr result was found in 18.4% of cases by periodogram analysis, but only in 0.2% of cases by PDM analysis. The figure suggests a reason: the 7.73 hr light curve approaches a perfect sinusoid more closely than the 7.77 hr version. Since there is no reason why the PDM analysis would miss a valid period in 99.8% of cases, it seems clear that the periodogram's frequent identification of the 7.73 hr period is due to the true light curve's having an imperfectly sinusoidal shape, thereby shifting some power away from the true 7.77 hr period in the periodogram analysis, but not in the PDM analysis since the latter has no problem with nonsinusoidal curves. This is consistent with Figure 7.

Figure 10 compares the 7.77 hr light curve with one from the 11.24 hr period preferred by Ortiz et al. (2007). There is no evidence for this period in our data, and the fit, as expected, is very poor. We have already noted that 7.8 and 11.41 hr peaks indicative of the true period and its alias exist in the periodogram of Ortiz et al. (2007).

Figure 11 compares the 7.77 hr result against a light curve folded at the strange period of 11.69 hr which was stronger than the 11.41 hr alias in the PDM analysis. The jagged appearance of the smoothed curve, together with a striking disagreement between the February 19 and February 26 data, clearly indicates that this period is unphysical. The specific sequence of gaps in our data set has evidently permitted this odd folding to have a formally low rms but it clearly cannot represent a real period. This is supported by the observation in Figure 6 that the 11.69 hr peak appears even in the data that simulate a perfect 7.77 hr sinusoid.

Finally, Figure 12 explores the possibility that the true period might be twice our preferred value of 7.77 hr: that is, that the light curve is double peaked. Folding with this period produces a clean, low-scatter light curve, as indeed it must, but the degree of symmetry is improbable. The twin peaks are very similar and spaced almost exactly 0.5 periods apart. While it is impossible absolutely to rule out a remarkably symmetrical albedo distribution that could produce such a double-peaked light curve, it is far more likely that 7.77 hr is the true period and the light curve is single peaked.

Finally, we conclude at the 95% confidence level that the true period is near 7.77 hr. Based on the PDM results from our Monte Carlo simulation, we quote a value of 7.7710 ±  0.0030 hr, where the uncertainty range encloses the 90% confidence interval.

Based again on our smoothed PDM fits from the Monte Carlo simulation, the amplitude of Makemake's rotational light curve is 0.0286 ±  0.0016 mag, where again the error bar corresponds to a 90% confidence interval. The amplitude of a sinusoidal fit to the light curve with a period of 7.77 hr is 0.0295 mag, easily consistent to within the error bars, which is not surprising since the PDM light curve still closely resembles a sinusoid.

The rotation period we have determined for 136472 Makemake is in the range of typical rotation periods for KBOs. There is no evidence for unusual tidal evolution like that of Pluto, whose rotation is tidally slowed by interaction with its moon Charon, nor for a massive collision like the one that appears to be responsible for Haumea's rapid spin (Brown et al. 2007b; Ragozzine & Brown 2007). This is consistent with Makemake's apparent lack of a substantial satellite (Brown et al. 2006b), and with the absence of any evidence for a large impact, in sharp contrast to Haumea.

Somewhat more remarkable is Makemake's very low photometric amplitude, which is barely a tenth that of Pluto, despite very strong similarities between the two bodies in other respects.

The spectrum of Makemake is nearly identical to that of Pluto, except that the former shows stronger absorption from methane ice. In particular, Makemake shares Pluto's red color at visible wavelengths (Licandro et al. 2006b, 2006a; Tegler et al. 2007; Brown et al. 2007a), in contrast to Eris, which shows strong methane absorption like Makemake, but has a neutral color (Brown et al. 2005). The picture presented by spectra is very consistent: Makemake is red, like Pluto, but Eris has a more neutral color (see especially, Figure 1 of Licandro et al. 2006a). However, some of the broadband photometry in the literature makes it appear that Eris and Makemake have identical colors. According to Rabinowitz et al. (2007), the B−V colors of Makemake, Eris, and Haumea are 0.828 ±  0.022 and 0.805 ±  0.015, and 0.646 ±  0.015, while Buratti et al. (2003) gives B−V = 0.82 ±  0.03 for Pluto. All this is consistent with other sources, except the color of Eris. According to Brown et al. (2005), the B−V color of Eris is 0.71 ±  0.02, which would make Eris much bluer than Makemake or Pluto, which would then stand out as resembling one another, just as they do in optical spectra. We suggest that though most of the Rabinowitz et al. (2007) photometry is excellent, there may be an error in the B magnitude for Eris, resulting in an incorrect color. This is suggested by their B-band phase slope for Eris, which is negative: an unphysical result. Taking the Brown et al. (2005) value rather than the Rabinowitz et al. (2007) value for the color of Eris resolves the apparent disagreement between the spectra and broadband colors: Makemake and Pluto have very similar colors; Eris is bluer than either.

In terms of albedo, Makemake is midway between Eris and Pluto. Eris has a V-band albedo of 0.86 ±  0.07 (Brown et al. 2006a), while that of Makemake is 0.7 ±  0.1 (Brown et al. 2007a), and that of Pluto is 0.52 ±  0.03 (Buratti et al. 2003).

The peak-to-valley amplitude of Pluto's light curve is 0.30, 0.26, and 0.21 mag in the B, V, and R bands, respectively (Buratti et al. 2003). The usual explanation for the red color and large photometric amplitude of Pluto is that it has distinct regions covered with dark red tholins which are generated when methane ice is bombarded with UV and ion radiation (Buratti et al. 2003; Brown et al. 2005). This also explains why the amplitude is lower at red wavelengths: the tholins are less dark relative to the surrounding bright ice in the R band than in the B band.

Because of Makemake's spectral resemblance to Pluto, and in particular its red color, the most obvious a priori assumption would be that it should have similar photometric amplitude. Instead, we find its peak-to-valley amplitude to be approximately nine times smaller, consistent with the results of Ortiz et al. (2007). This is a truly remarkable difference, suggesting either that Makemake's surface is very uniform and unlike Pluto's, or that we see it with a nearly pole-on viewing geometry. Brown et al. (2005) suggest that Eris' neutral color may be due to its larger heliocentric distance (about 90 AU), at which temperatures are sufficiently low for the red tholins to be covered up with a planet-wide frost, which would consist of methane possibly mixed with nitrogen. This would also explain the object's low photometric variability and high albedo. Ortiz et al. (2007) apply the same global frost theory to explain the low photometric amplitude of Makemake (current heliocentric distance about 52 AU, still considerably greater than Pluto's), but this explanation seems inconsistent with Makemake's red, Pluto-like color. Adding to the complication is Makemake's higher-than-Pluto albedo, which may suggest it is "partially frosted," and indications that methane ice on Makemake may be transparent to centimeter depths (Brown et al. 2007a), so that what is below would not necessarily be wholly obscured.

Ortiz et al. (2007) note that the likelihood of both Eris and Makemake being oriented pole-on to us is very low, and argue that most likely there is a common physical reason for the low photometric variability of both objects, i.e., some type of global frost scenario. However, the color difference between Makemake and Eris may present a problem with explaining the low photometric variability of both by the same type of global frost.

We suggest that instead of a global frost, Makemake's low photometric amplitude may be due to a nearly pole-on viewing geometry. If this is the case, the light-curve amplitude and shape will change relatively rapidly as the object moves in its orbit. Under this supposition, the light curve should look substantially different in 20 years, while if the surface is truly very uniform, the light curve will remain essentially unchanged.

A less definitive but still interesting test that could be performed on a shorter timescale is the measurement of Makemake's light-curve amplitude at different wavelengths. If the object has Pluto-like dark red patches and is simply being viewed pole-on, the light-curve amplitude should vary with wavelength just as Pluto's does. Scaling from Pluto's wavelength dependent amplitudes and our own value of 0.0286 ±  0.0016 mag for Makemake's peak-to-valley V-band photometric amplitude, we would predict a B-band amplitude of 0.0330 mag and an R-band amplitude of 0.0231 mag. This difference could easily be detected with photometry as precise as that we have presented herein. If the B-band light curve did not show a markedly larger amplitude than the R band, the hypothesis that Makemake is a pole-on Pluto analog could be dismissed. We note that Ortiz et al. (2007) determined an R-band light-curve amplitude of 0.03 ± 0.01 mag for Makemake, but given the large uncertainty, this does not rule out the value we predict for a pole-on Pluto scenario.

We have obtained some of the most precise time-series photometry ever acquired for a KBO. A similar project performed on Eris might yield the first confident determination of its rotation period as well. It is even possible that similar work could be extended to a precise measurement of Eris' light-curve amplitudes in the B, V, and R bands. This would allow a test of the theory that Eris' red tholins are covered up with a global methane frost: if they are, the light-curve amplitude should be the same at all wavelength bands.

We have obtained precise, well sampled time-series photometry of the dwarf planet 136472 Makemake, formerly known as 2005 FY9, and have determined accurate values for its rotation period and rotational light-curve amplitude. Its rotation period is 7.7710 ±  0.0030 hr, with 90% confidence. The amplitude of its rotational light curve is 0.0286 ±  0.0016 mag in the V band.

Makemake's rotation period is in the typical range for KBOs, unlike those of Pluto and Haumea, which have been dramatically altered (by tidal evolution in the case of Pluto, and probably by a giant impact in the case of Makemake; Brown et al. 2007b).

The very low photometric amplitude of Makemake is surprising given the much larger variability of Pluto, and the spectral similarity between the two objects. Makemake may have an extremely uniform surface, but the previous suggestion of a global methane frost due to the low temperatures encountered at large heliocentric distances appears inconsistent with the object's Pluto-like red color. Our observations are also consistent with a Pluto-like variegated surface combined with a nearly pole-on viewing geometry.

We have demonstrated that, given a 1.5 m telescope and a high-quality CCD in good weather, very careful data reduction can yield considerably more precise relative photometry of bright KBOs than has generally been published in the past. Similar techniques applied to Eris might well result in a confident determination of its rotation period, which has so far been elusive. Observing Makemake itself at different wavelength bands could determine whether or not its light curve is dominated by Pluto-like deposits of dark red tholins—if so, the photometric amplitude should be lower in the R band and greater at B. The same type of extremely precise multi-wavelength observations applied to Eris could test the theory that the red deposits that probably exist on its surface are covered with a uniform methane frost—if so, the light-curve amplitude should be wavelength independent. Time-series photometry of the precision we present here represents a potentially powerful tool for the study of large objects in the Kuiper Belt.

This research has made use of the SIMBAD online database, operated at CDS, Strasbourg, France.

This research has made use of the VizieR online database (see Ochsenbein et al. 2000).

This research has made extensive use of information and code from Press et al. (1992).

Facilities:61'' Kuiper