Issue 
A&A
Volume 672, April 2023



Article Number  A61  
Number of page(s)  6  
Section  Celestial mechanics and astrometry  
DOI  https://doi.org/10.1051/00046361/202244816  
Published online  29 March 2023 
CCD astrometric measurements for the dwarf planet Haumea by groundbased telescopes
^{1}
Department of Computer Science, Jinan University,
Guangzhou
510632, PR China
email: tpengqy@jnu.edu.cn
^{2}
IMCCE, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Université,
Université de Lille,
75014
Paris, France
^{3}
Shanghai Astronomical Observatory, Chinese Academy of Sciences,
Nandan Road 80,
Shanghai
200030, PR China
^{4}
SinoFrench Joint Laboratory for Astrometry, Dynamics and Space Science, Jinan University,
Guangzhou
510632, PR China
Received:
26
August
2022
Accepted:
15
February
2023
Aims. The purpose of this work is to investigate an astrometric method to accurately measure the precise position of the center of mass of the dwarf planet Haumea and its satellites, based on the unresolved images taken by groundbased telescopes.
Methods. A total of 29 nights of CCD observations are reported of the dwarf planet Haumea, taken during the period from February 7 to May 25, 2022 by three telescopes located in Yunnan Province, China. The pixel centroid of the unresolved image of Haumea and its satellites is measured with respect to its reference stars in the Gaia EDR3 star catalog. The new theoretical position of Haumea (the center of mass of Haumea and its satellites is assumed) is computed by the Jet Propulsion Laboratory (JPL) Horizon System. The observed minus computed (O–C) positional residuals of Haumea in right ascension and in declination show a significant and periodic variation. Furthermore, when the ephemeris position of Hi’iaka, the larger and brighter satellite of Haumea, with respect to Haumea itself is taken from the Miriade ephemeris service of the Institut de mécanique céleste et de calcul des éphémérides (IMCCE) website, the observed positional measurements of the center of mass of Haumea and Hi’iaka are derived precisely after using a new method.
Results. The (O–C) positional residuals are confirmed to be caused mainly by the motion of Hi’iaka around its primary. The final mean (O–C) residuals of Haumea and its satellites are found with respect to the most recent version of the JPL ephemeris. As a byproduct, a parameter that connects the mass ratio to the light ratio of Haumea and Hi’iaka (by Eq. (2) in the text), is introduced to derive the final mean (O–C) residuals.
Key words: astrometry / methods: data analysis / techniques: image processing / Kuiper belt objects: individual: Haumea
© The Authors 2023
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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1 Introduction
The astrometry of a transNeptunian object (TNO) is of importance to the construction of its ephemeris and to study its dynamical evolution. Based on its refined ephemeris, the stellar occultation by a TNO can be predicted accurately. However, a precise astrometry presents several challenges for a faint TNO. Although the Gaia DR2 catalog, or the Gaia DR3 star catalog, can provide a large enough number of stars with accurate positions even for a small field of view, we must have a largediameter telescope to get enough signaltonoise ratio (S/N) for a faint TNO. In addition, the differential color refraction in the atmosphere causes a systematic error. According to Stone (2002) and Lin et al. (2020), a strategy to minimize this effect is to make observations near transit or to use filters, ideally nearIR filters. However, the former approach would limit our observational zenith distances, and the latter approach usually gives rise to a notorious fringe pattern in the CCD frame, which has to be treated carefully. In addition, binary TNOs or TNOs with satellites also require special treatment since their centroid will not lie on the center of the primary, but between the primary and satellite, and hence will produce a systematic difference. It is after overcoming these difficulties that an accurate stellar occultation prediction by a TNO is possible. We refer to the stellar occultations in Elliot et al. (2010), Sicardy et al. (2011), Ortiz et al. (2012, 2017, 2020), BragaRibas et al. (2013), BenedettiRossi et al. (2016, 2019), and Schindler et al. (2017) for examples.
A few big TNOs have one or two satellites, especially for dwarf planets (e.g., Haumea, Makemake and Eris). Astrometry of these dwarf planets by a conventional groundbased telescope (rather than an adaptive optics telescope) can also reveal valuable information on the orbital motion of their satellites, and can be used to study the stability of a dwarf and its satellite (such as tidal locking).
Ortiz et al. (2011) used a 45 cm f/2.8 remotely controlled telescope with a large format CCD camera of 4008 × 2672 pixels for the astrometry of a big TNO (90482) Orcus. Specifically, their telescope was pointed at the same field of sky and many CCD images of the target were taken over 18 nights during the period of 33 days. After the observed positions of Orcus with respect to the same reference stars were compared with the Jet Propulsion Laboratory (JPL) ephemeris, a significant and periodic signal appeared. The period was 9.7 days by measurement. At the same time, Ortiz et al. (2011) took the relative photometry of the target also with respect to the same reference stars (as those for the astrometry) and found the same period signal in the light curve. All these revealed that Orcus has a satellite, and the period of the satellite’s rotation around its primary is equal to its spinning period. In other words, the satellite is in the state of tidal locking.
In order to predict accurately stellar occultations by the dwarf planet Haumea, Ortiz et al. (2017) used a 77 cm f/3 telescope with a 4096 × 4096 pixel CCD camera to do astrometry of Haumea. Their observations were taken during the period from April 1 to July 4, 2016. After the observed positions of Haumea were compared with those drawn from the most recent version of the JPL ephemeris, they found that the residuals in declination of Haumea significantly showed a periodic signal with a period of 49.5 days, although no periodic signal was found in right ascension. The period of 49.5 days was coincident with the orbital period of Hi’iaka, the larger and brighter satellite of Haumea. Furthermore, when the Miriade ephemeris service of IMCCE for the satellite Hi’iaka was referred to its primary, the positional residuals of Haumea were found to be closely correlated with the positional difference of the satellite with respect to its primary Haumea. All this confirmed that the positional residuals of Haumea revealed the orbital motion of Hi’iaka.
A very recent work of Gaia (Tanga et al. 2022) also showed a similar valuable astrometric signal for an asteroid with a satellite. The orbital motion of an asteroid with a satellite in its ephemeris reflected only their center of mass motion, and the observed positional residuals of the asteroid exhibited the motion of their photocenter with respect to their center of mass. This relative motion was called wobbling or oscillating. Specifically, Gaia could successfully detect the wobble of the main belt asteroid (4337) Arecibo and its satellite. The magnitude of the wobble was found to be about 0.8 milliarcsecond (mas), (Tanga et al. 2022).
In this paper, we report on the three moderate telescopes located in Yunnan Province, China, that were used to take CCD observations of the dwarf planet Haumea. We aimed to measure precisely the position of the dwarf planet and to confirm Hi’iaka’s motion in the positional residuals when its observational positions were compared with the most recent version of the JPL ephemeris. Moreover, the observed positional offsets of the center of mass of Haumea and Hi’iaka from their theoretical center of mass was derived via a new reduction method.
The remainder of this paper is organized as follows. Section 2 explains in detail our observations. Section 3 deals with data reduction and analysis, including image processing, the residual solution for Haumea, residual analysis for its period, and the parameter solution when compared with the ephemeris of Hi’iaka. Finally, some discussions and conclusions will be drawn in the last section.
2 Observations
Our observations consist of 463 CCD frames for Haumea taken with three telescopes during the period from February 7 to May 25, 2022. A total of 33 observations were made at three sites over 29 nights. More details of the instruments used and image characteristics are described in Table 1.
All observations were taken through the Johnson I filter, and the observational zenith distances were less than 60°. According to Stone (2002) and Lin et al. (2020), such observations by the filter rendered differential color refraction negligible in most astrometric applications. However, CCD frames taken with the I filter often presented some notorious fringes (the removal of the fringes is explained in Sect. 3.1), especially for our observations taken with the 0.8 m telescope. The exposure times ranged from 300 s to 600 s depending on the different telescopes and different weather conditions. In addition, bias and flat CCD frames were also taken. For the CCD frames by the 0.8 m telescope, five dark frames on each night have also been taken after target observing since March 3, 2022.
3 Reduction and analysis
3.1 Image processing
All CCD frames were preprocessed by conventional bias and flatfield corrections. For the CCD frames by the 0.8 m telescope, a dark correction was also done for each night’s science frame. Furthermore, a pattern of fringes occurred for the observations acquired by the 0.8 m telescope, due to the I filter used. The fringe removal was made after bias, flat, and dark corrections (for more details, see Thirouin 2013).
After these preprocessed operations and the fringe removal, the pixel positions of all the images of the stars and (unresolved) of Haumea in each CCD frame were determined with some centroiding algorithms. Although a point spread function (PSF) is more suitable for measuring a precise pixel position of a star or our target image, we discovered that the twodimensional Gaussian fit centering algorithm is very flexible and a good enough algorithm for our groundbased CCD images (Lin et al. 2021; Guo et al. 2022). Then, we recognized catalog stars and matched their pixel positions with the star catalog Gaia EDR3 (Gaia Collaboration 2021).
3.2 Observed minus computed (O–C) positional residuals of Haumea
Because of a long exposure time and a quite wide field of view (at least 9′ × 9′) for each telescope (see Table 1), we usually have at least 40 stars in the star catalog of Gaia EDR3 (Gaia Collaboration 2021). The faintest stars can reach about 20.5 Gmagnitude for the 0.8 m telescope with an exposure time 360 s, and 20.7 Gmagnitude for the 2.4 m telescope with an exposure time 100 s, and about 20.0 Gmagnitude for the 1 m telescope with an exposure time 500 s because of quite bright sky background due to its nearness to the city. According to our experience (Peng & Fan 2010; Lin et al. 2019; Guo et al. 2022), we can adopt a 30constant plate model (i.e., a fourthorder polynomial in each direction) to connect the pixel position of each star and its standard coordinate after the central projection (Green 1985) in each CCD field of view. Moreover, we improve the plate model solution by a weighted strategy (see Lin et al. 2019, 2020 for details of the practice). After the plate model solution, the pixel position of Haumea in each CCD field can be precisely transformed to its standard coordinate and can be compared with its theoretical position (i.e., topocentric and astrometric position) in the most recent version of the JPL ephemeris^{1}. Thus, we can derive the observed minus computed (O–C) positional residual of Haumea in each field of view. Because of its relative dimness (its apparent magnitude is about 17.5), the dispersion of each residual is somewhat great (usually about 0.″01 ~0.″04 for an individual observation, dependent on seeing, S/N, and telescope used). We average these residuals for each night’s observations in right ascension and declination, respectively, and thus deliver their uncertainty in each direction as well as the correlation coefficient. The mean (O–C) residuals, hereafter called raw residuals, in each direction and their uncertainties (i.e., the estimates of errors) and the correlation coefficients are discussed further in Sect. 3.4.
Instrumental details for telescopes and CCD cameras.
Fig. 1 Lomb–Scargle periodograms of the residuals in right ascension and declination. The left panel is for right ascension and the right panel for declination. The wide spike frequency of about 0.02 cycles/day (i.e., about 50 days for the period) in each panel can be notably found since it is the maximum spectral power. 
3.3 Periodic variations in positional residuals
In order to find out the periods in the raw positional residuals of Haumea, a LombScargle periodogram analysis (Lomb 1976; Scargle 1982) is made. In detail, we performed the time correction for a light travel for the mean time (Julian date). After this the time series in right ascension and declination are analyzed according to the algorithm in Press et al. (1992); the results are shown in Fig. 1. It is obvious that a periodic signal with a frequency of about 0.02 cycles/day (i.e., about 50 days for the period) exists in each direction. Specifically, the periods are found to be 47.1 ± 1.4 days and 46.5 ± 0.9 days in right ascension and declination, respectively. Because of the relative long orbital period of Hi’iaka and the observing window of only about two periods, the solved period in each direction has quite large deviation according to VanderPlas (2018) with respect to the nominal orbital period, 49.462 days, of Hi’iaka according to Ragozzine & Brown (2009). Additional contributions to these deviations can be the light variation of the faint satellite of Haumea, Namaka, and rings. Obviously, more observations in the future are needed to investigate and reduce the differences.
On the other hand, we try to determine whether the raw positional residuals were correlated or not with computed theoretical positions of Hi’iaka around its primary according to the pioneering work of Ortiz et al. (2011). Here, we neglect the influence of Namaka, due to its low brightness and small mass as the practice by Ortiz et al. (2017). As such, we retrieve the relative position of Hi’iaka with respect to Haumea via the website of IMCCE^{2}. Figure 2 shows the strong correlations in each direction. In detail, the correlation coefficient is 0.81 in right ascension and 0.97 in declination.
In summary, the raw positional residuals of Haumea show its periodic variations with a period almost equal to the orbital period of Hi’iaka. In addition, these residuals have significant correlations with the positional differences of Hi’iaka, with respect to Haumea, in right ascension and declination. Therefore, our observed raw positional residuals reflect without doubt the orbital motion of Hi’iaka with respect to its primary, Haumea.
3.4 Further solution for the observations
We denote the mass ratio and the light ratio of Haumea and Hi’iaka as r_{M} and r_{L}, respectively: r_{M} = m_{H}/m_{S} and r_{L} = L_{H}/L_{S}, where m_{H} and m_{S} are the mass of Haumea and Hi’iaka, respectively. Similarly, L_{H} and L_{S} are the light of Haumea and Hi’iaka, respectively. According to the definitions of center of mass and photocenter, we can derive easily (see Fig. 3) the angular separation Δ of the photocenter with respect to the center of mass by following Eq. (1), (1)
where d is the angular separation of Hi’iaka and Haumea at some observational instant and whose two projections (x_{c}, y_{c}) in right ascension and in declination can be obtained from the satellite’s ephemeris for the same observational instant. In the derivation, we made an approximation that no phase effects are introduced so that the individual photocenter offsets for each component are neglected. A similar equation can be found in Tanga et al. (2022), where they assume spherical components of identical albedo and bulk density (see their Eq. (6)).
In order to express more clearly each quantity, we let (2)
Figure 3 shows us the schematic diagram of the satellite’s rotation around its primary, and some relations among quantities concerned.
For the system of Haumea and its satellites, it is well known that Haumea itself has rather great light variations due to its elongated shape (Ortiz et al. 2017) and that Hi’iaka also has its light variation (Ragozzine & Brown 2009). According to Brown et al. (2006), Hi’iaka has a fractional brightness of 5.9 % with respect to Haumea. As such, the light ratio r_{L} is estimated at ~16.9. In addition, according to Ragozzine & Brown (2009), the mass ratio r_{M} of Haumea and Hi’iaka is estimated at 223.8. Therefore, the value k in Eq. (2) is about 0.051. Taking into account the light variation of Haumea and Hi’iaka, k will show oscillations on the order of 0.008 according to Rabinowitz et al. (2006) in which the observed magnitude of Haumea (and its satellites and ring) as a whole had a peaktopeak amplitude 0.28. In practice, we cannot depend on this estimated value since the light variations exist in its two faint satellites as well. In addition, we use the JohnsonI filter. A reasonable treatment for the variation is to ignore the variations and solve for a constant mean value for k, which is solved in our reduction. The solved k reflects the difference between the mass ratio and the light ratio of Haumea and Hi’iaka. If the difference is zero, our observational position of Haumea’s image directly reflects the position of the center of mass (i.e., we cannot find any evidence of a satellite in our observations). Otherwise, we can measure the difference. Specifically, if the light variations of Haumea and its satellite are determined by some technique, we can measure the mass ratio of Haumea and Hi’iaka. We refer to Foust et al. (1997) for an example; they show the mass ratio of Pluto and its satellite Charon.
When our observations are compared with the ephemeris of Haumea via JPL website^{3} (i.e., JPL#108 for Haumea and DE441 for other planets), the raw positional residuals (O–C) should contain both the positional difference (Δ) of the photocenter relative to the center of mass and a constant offset (A, D), which reflects the difference of observed center of mass with respect to the ephemeris center of mass for Haumea and Hi’iaka (see Fig. 3).
If we make a projection of the angular separation Δ in right ascension and declination (Δ_{x}, Δ_{y}), and take the constant offsets (A, D) mentioned above into account as well, we have (3)
where i = 1,2,3,…, N; N = 33 is the number of mean observations; (X^{i}, Y^{i}) are the observed raw positional residuals in right ascension and declination, respectively, at epoch i; (, ) are the ephemeris positional differences between Hi’iaka and Haumea in right ascension and declination, respectively, at epoch i; and (, ) are the final residuals in right ascension and declination, respectively, at epoch i. We fitted all our raw positional residuals (X^{i}, Y^{i}) by minimizing a χ^{2} function (i.e., a threeparameter model): (5)
where σ_{X,i}, σ_{Y,i} is the uncertainty on the position averaged over each night in right ascension and declination, respectively, and ρ_{i} is the correlation coefficient of observations, at epoch i. These three quantities can be estimated after each night’s observations. The quantity det(Q_{i}) is the determinant of covariance matrix Q_{i.} After the solution we obtain the estimated three parameters with quite small uncertainties in Table 2. Specifically, the derived value of parameter k is the same whether the correlation coefficient is taken into account or not. For the other two parameters (A, D), their derived values have very small differences (not greater than 0.002 arcsec) whether the correlation coefficient is taken into account or not.
We should note that the solved offsets (Â, ) reflect the difference between the measured positions of the center of mass and the positions of center of mass from the ephemeris for Haumea and its satellites. In addition, the solved mean value of is consistent with its nominal value in the estimated variation range. Figure 4 shows the fitted results of the positional residuals of Haumea by Eq. (5). On the whole, our fitted results show a quite good agreement with the raw observed residuals, especially in declination. Figure 5 shows the two kinds of residuals, where the final positional residuals after the threeparameter model solution become much smaller.
Fig. 2 Correlation of the residuals of Haumea with the positional difference of Hi’iaka with respect to its primary, Haumea, according to the ephemeris of IMCCE. The left panel shows the correlation in right ascension and the right panel the correlation in declination. Significant correlation appears in each direction. 
Fig. 3 Schematic diagram. In left panel, the primary (P, here Haumea) and the secondary (S, here Hi’iaka) rotate mutually around their center of mass C_{M} , but only their photocenter C_{L} is visible and measurable. Thus our observed trajectory of the photocenter will be its rotation motion around the center of mass C_{M} along an approximate ellipse when the center of mass is chosen as a reference origin. The angular separation of the secondary relative to its primary is supposed to be d (it is usually less than the atmospheric seeing for a groundbased telescope, thus only an unresolved image is observed) and its projections to right ascension and declination are (x_{c}, y_{c}), respectively. Correspondingly the separation between the photocenter and the center of mass is Δ. In right panel, the center of mass in the ephemeris for the primary and its secondary component is also shown as the reference origin. Here, the true center of mass C_{M} has an offset (A, D) from . In addition, an ellipse is drawn for the trajectory of the photocenter C_{L} going around the center of mass C_{M}. The projections of the angle separation Δ in right ascension and declination are Δ_{x} and Δ_{y}, respectively. 
Two sets of parameters (K, A, D) estimated without (Solution A) and with (Solution B) the consideration of the correlation coefficient of observed right ascension and declination at each epoch.
4 Discussion and conclusion
It is confirmed from our observations that a moderate (about 1 m aperture) groundbased telescope can deliver a very small positional error for the center of mass of a dwarf planet and its satellite. This success may be mostly due to the highly accurate star catalog Gaia EDR3. In addition, some techniques and conditions are also important to acquire accurate observational positions of Haumea. First, the filter used (i.e., the JohnsonI filter) can effectively avoid the effect of differential color refraction (DCR) on our target’s positional measurement. However, its sideeffect, the induced fringe, needs to be carefully removed because of its positional effect, especially on a faint target or a faint reference star. Second, a weight strategy is an easily neglected technique in our astrometric reduction. However, it is quite significant to the calibration of the CCD field of view, and hence to our target’s positional measurement. Third, the new reduction method used in this work is to fit our raw positional residuals by the threeparameter model via Eq. (5). The solution for the model allows us to obtain an adjustable parameter k as well as the positional offsets (A, D) of the observed center of mass relative to their ephemeris’ center of mass of Haumea and its satellite. The parameter k is is noteworthy for its direct relation with the light ratio and the mass ratio of the primary and its satellites via Eq. (2). In a strict sense, we might need to improve the positional effect from the light variations of Haumea and Hi’iaka. In this way, the solved parameter can accurately reveal the true mean light difference of Haumea and Hi’iaka. However, the effect may be very limited in this work since the average raw residuals are obtained from one night’s observations. Finally, a different atmospheric seeing might cause some error in the positional measurement for our target when Hi’iaka has a rather large angle separation, according to some careful considerations by BenedettiRossi et al. (2014), although they refer to Pluto and its major satellite Charon. In the future work, we will take other positional effects (e.g., rings and other satellites) into account, and focus on the mass ratio of Haumea and Hi’iaka. In addition, we will also determine an accurate orbital period of Hi’iaka when longterm and accurate observations are obtained.
Fig. 4 Raw positional residuals of Haumea. A 1σ error bar is fitted by a function of the χ^{2} in Eq. (5) when the ephemeris of satellite Hi’iaka is referred to and without considering the correlation coefficient. The left panel is the fitted results in RA and the right panel in DE. 
Fig. 5 Two kinds of residuals. The dark squares are the raw positional residuals and the red circles are the final positional residuals after the threeparameter model solution in Eq. (5) without considering the correlation coefficient. 
Acknowledgements
This work was supported by the key special project of the Ministry of National Science and Technology (grant no. 2022YFE0116800), by the National Natural Science Foundation of China (grant nos. 11873026, 11273014), by the Joint Research Fund in Astronomy (grant no. U1431227) under cooperative agreement between the National Natural Science Foundation of China (NSFC) and Chinese Academy of Sciences (CAS), and by the China Manned Space Project with no. CMSCSST2021B08. This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC; https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. We acknowledge the support of the staff of the Lijiang 2.4 m telescope. Funding for the telescope has been provided by Chinese Academy of Sciences and the People’s Government of Yunnan Province. In addition, we are grateful for Drs F. Li, Y. Yuan, and other staff for the group of 0.8 m telescope, Dr Ε. G. Zhao for 1 m telescope to take observations and providing us help of obtaining observations. We thank Prof Υ Ν. Fu from the Purple Mountain Observatory, and Prof S. B. Qian from the Yunnan Observatory to give us great support and helpful discussions. We also thank D. Li, X. Chen, J. N. Hao, Y. Y. Hong and J. L. Cao who help us match and process the observations.
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All Tables
Two sets of parameters (K, A, D) estimated without (Solution A) and with (Solution B) the consideration of the correlation coefficient of observed right ascension and declination at each epoch.
All Figures
Fig. 1 Lomb–Scargle periodograms of the residuals in right ascension and declination. The left panel is for right ascension and the right panel for declination. The wide spike frequency of about 0.02 cycles/day (i.e., about 50 days for the period) in each panel can be notably found since it is the maximum spectral power. 

In the text 
Fig. 2 Correlation of the residuals of Haumea with the positional difference of Hi’iaka with respect to its primary, Haumea, according to the ephemeris of IMCCE. The left panel shows the correlation in right ascension and the right panel the correlation in declination. Significant correlation appears in each direction. 

In the text 
Fig. 3 Schematic diagram. In left panel, the primary (P, here Haumea) and the secondary (S, here Hi’iaka) rotate mutually around their center of mass C_{M} , but only their photocenter C_{L} is visible and measurable. Thus our observed trajectory of the photocenter will be its rotation motion around the center of mass C_{M} along an approximate ellipse when the center of mass is chosen as a reference origin. The angular separation of the secondary relative to its primary is supposed to be d (it is usually less than the atmospheric seeing for a groundbased telescope, thus only an unresolved image is observed) and its projections to right ascension and declination are (x_{c}, y_{c}), respectively. Correspondingly the separation between the photocenter and the center of mass is Δ. In right panel, the center of mass in the ephemeris for the primary and its secondary component is also shown as the reference origin. Here, the true center of mass C_{M} has an offset (A, D) from . In addition, an ellipse is drawn for the trajectory of the photocenter C_{L} going around the center of mass C_{M}. The projections of the angle separation Δ in right ascension and declination are Δ_{x} and Δ_{y}, respectively. 

In the text 
Fig. 4 Raw positional residuals of Haumea. A 1σ error bar is fitted by a function of the χ^{2} in Eq. (5) when the ephemeris of satellite Hi’iaka is referred to and without considering the correlation coefficient. The left panel is the fitted results in RA and the right panel in DE. 

In the text 
Fig. 5 Two kinds of residuals. The dark squares are the raw positional residuals and the red circles are the final positional residuals after the threeparameter model solution in Eq. (5) without considering the correlation coefficient. 

In the text 
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