Light curve analysis of main belt asteroids 4747, 5255, 11411, 15433, 17866

1 Introduction

The investigation of asteroid properties is important for the development of the evolution theory of the solar system and the classification of small solar system objects. Because some of these objects can collide with the Earth, asteroids are also important for having significantly modified the Earth’s biosphere in the past. They will continue to do so in the future. This might sound surprising because asteroids are considered a nuisance due to their potential to impact Earth and trigger mass extinctions. But an emerging view proposes that asteroid collisions with planets may provide a boost to the birth and evolution of complex life (Castillo and Vance 2008) and (Houtkooper 2011). During the early solar system, the carbon-based molecules and some heavy elements that served as the building blocks of life may have been brought to the Earth via asteroid and comet impacts. Asteroid studies will allow us to answer the ambiguous question about the origins of life on Earth. On the other hand, the next step in the human exploration and exploitation of space will be highly dependent on extracting materials (primarily water and minerals) from space sources. It is highly probable that the success and viability of human expansion into space will depend on the ability to exploit asteroid resources. Therefore, a detailed physical and compositional assessment of the population will be required during the next decade before human missions are sent to these objects. The photometric study of light curves can obtain additional information about size, rotation period, the structure of objects, and the existence of craters and ice fields on the surface, which is very important data for space missions. Asteroids shine due to the Sun’s light reflecting on their surface and depend on surface albedo (from surface characteristics: chemical composition; regoliths which cover the object). If an asteroid is not spherical, its brightness might vary due to one or more of the following factors: the asteroid’s distance to the observer and to the Sun is changing; the asteroid’s phase is changing (just like the Moon’s). All of the above plus the shape of the asteroid and its periodic rotation as well as the precession of the axis of rotation are reflected in changes in brightness over time – which astronomers understand under the term light curve of an asteroid. The importance of light curve acquisition and the complexity of the analysis at the same time can be understood from this.

2 Observations

At the Baldone Astrophysical Observatory (IAU Code 069), astronomers operate with a Schmidt-type 1.2 m telescope installed with two STX-16803 charge-coupled devices (CCDs). The brightness limit in the visual range of the telescope without a filter is 22 magnitude at night with good transparency and calm images. CCD parameters are quantum effectivity of about 80%, the size of one pixel is 9 × 9 microns, and linear size 4,096 × 4,096 pixels, which corresponds to 53 × 53 arcmin of the field of view. Monitoring of asteroids in the Baldone observatory took place from 2008 mainly without a filter. Part of the clear nights in the last 3 years is devoted to the studying dynamics of main belt asteroids in the G(RP) passband. Observations also managed to use nights with a small phase of the Moon. The list of observable asteroids was compiled using the links of the Minor Planet Center NEO checker (MPC 2022) and MPC light curve database (ALCDEF 2022). The list included those NEO and main belt asteroids with a brightness greater than 18 magnitudes without period data. Observations of selected asteroids in Baldone Observatory are usually made on three to five following nights. Three- to five-hour long series of observations are dedicated to each asteroid at night. On average, it gives more than a hundred observations for each object. The particular asteroid observations were made in 2020–2021 mainly with exposures of 180 or 240 s, to achieve a signal-to-noise ratio greater than 20. The details of observational circumstances at Baldone observatory are given in Table 1. The table contains observation dates, exposition, asteroid phase, and distance from the observer (Delta) and Sun (R) in astronomical units (AU).

3 Light curves analysis

The G(RP) magnitudes for reference stars were taken from the GAIA DR2 release (Brown et al. 2018). Usually, 5–6 reference stars: with colors close to the Sun, some brighter and some dimmer, than an asteroid, were selected for the processing of one series of observations. The images were calibrated and measured using Maxim DL software. Measurement of magnitudes of objects was made after the application of standard procedures of master flat and master dark images. For further processing, we selected only that series where the reference star’s brightness errors at an average are smaller than 0.03 magnitudes. It helps to discard observations with poor sky instant transparency. Each measurement of an asteroid consists of a time and apparent magnitude couple. Both values must be corrected for each measurement series because the distance of an asteroid relative to Earth and to the Sun changes:

where D is the asteroid’s distance from the observer. The magnitudes of all series are corrected depending on the distance from the observer and from the Sun (Zeigler and Hanshaw 2016):

where D is the asteroid distance from the observer and R is the distance to the Sun and R 0 and D 0 is the same for the first observation in the first series. All subsequent series magnitude measurements are reduced to the magnitudes of the first series of observations by correction:

where k is the slope coefficient of the phase diagram for phases in intervals 10–30°, and Ph i and Ph 0 are i and the first phase series, respectively. We thus use linear regression to fit the lightcurve amplitude-corrected data. The individual, average phase slope for particular asteroids is given in Table 2. The error of the slope coefficient is obtained by comparing the phase diagrams of different observatories. After all, corrections, to derive asteroid periods from their lightcurves, we solved Eq. (4) by least squares with the assumed period P (Pravec and Harris 2000):

where V is the average brightness for a single light curve, Ak and Bk are the Fourier series coefficients, P is the synodic period, t is the time of observation (in Julian days), and t 0 is the time of the first observation. At each step, a r 2 value was derived and a global minimum was located. To estimate the uncertainty of the data points, we rescaled the smallest r 2 per degree of freedom to unity. A detailed description of the algorithm is presented by Kwiatkowski et al. (2009). Classic Fourier series (F) method gives usable results analyzing long series observation in multiple following nights when the rotation period is not longer than 7–10 h. Previous attempts to obtain asteroid rotation periods using only the F series method in the analysis did not always lead to convincing results, for example, for asteroid 1999 XC221 (Eglitis et al. 2022). The Fourier method is very sensitive to gaps in observations, especially when summarizing data from small series of observations, as well as data from different oppositions, where are large shifts in the brightness range. In this situation, the Lomb–Scargle (L–S) periodogram is a more reliable method used to analyze unevenly sampled time series data for periodic signals. It is especially valuable when dealing with nonuniformly spaced or sparsed data just like the rest of these methods (VanderPlas 2018). The Lomb-Scargle method takes the classic periodogram and generalizes it further and includes two coefficients in front of the trig function terms and adds a time-shifting term to allow for unevenly spaced data. In data processing with the L–S periodogram method, it is very important to correctly choose the frequency step width with which we pass the entire frequency range under study, too fine a grid can lead to unnecessarily long computation times that can add up quickly in the case of large surveys, while too coarse a grid risks entirely missing narrow peaks that fall between grid points. The step can be calculated by the following formula:

where recommended values of n are usually between 5 and 10; T is a common observation range in days. We used n = 10 in our study. The fitting quality in L–S method of the sinusoidal model is characterized by χ 2 . The closer χ 2 is to one, the better the agreement of the model with the observational data. One drawback of this method is that it only works if the white noise being sifted through is uncorrelated, and otherwise, the results will not be great. Θ is a term that compares the population variance against the data variance, and this is the term that is minimized. (VanderPlas 2018). The third-phase dispersion minimization (PDM) method is valuable when dealing with nonuniformly spaced data. The basic concept is that the observations are grouped into bins of roughly the same phase. The variance in each bin can now be calculated. The overall variance is the sum of the variance of the samples: data are cut for each test period until the total variance is minimized (specifically, it looks for local minima, not global minima). In our research, the bins from 5 to 10 were checked. Θ is a term that compares the population variance against the data variance, and this is the term that is minimized. The period is a good fit if it is close to 0, and if it is close to 1, it indicates a wrong period (Stellingwerf 1978). PDM has the biggest noticeable disadvantage, as comfortably more than 100 observations are needed in order for it to work well. Anything less can result in a wrong estimated period. The other two methods do not have this problem as much and so can be relied on for smaller samples of data. In cases where the results of F analysis indicated a period exceeding 7 h, published MPS Mauna Loa (IAU Code T08), Haleakala (IAU Code T05), and Ponte Uso (IAU Code L41) observatories and Catalina Sky Survey (IAU Code 703) brightness data were analyzed additionally. In addition, Punto Uso r and g brightness data as well as Mauna Loa and Haleakala o and c brightness data were analyzed separately. The true rotation period of the asteroid is considered to be the mean weighted period obtained by equation 8, using all calculated periods from data of different observatory measurements. When obtaining the rotation period in this way, the scatter around the mean light curve is taken into account by the following equation (using periodogram characteristics in Table 2: Θ for the PDM method and w = χ 2 or w = 1 − r 2 in the cases of L–S and F methods, respectively).

The number ( N ) of measurements is taken into account by the following equation:

The resulting mean weighted period is:

where P k periods (see P(L–S) and P(PDM) in columns five and seven of Table 2, respectively) are obtained with different methods using data from different observatories, N i is the number of data and W i is a character of the power spectrum, which gives a measure of the scatter around the mean light curve. Θ i also is a measure of the scatter around the mean light curve in the case of the PDM method. Y k is the corresponding power spectrum characteristic ( W k or Θ k ) of the period under consideration. The codes of observatories with passband index are in the second column, and the number of magnitude measurements, obtained in the span range of time (column four), is in the third column of Table 2.

4 Rotation periods

From the beginning period, diapason from 0.5 h till 100 h are analyzed. Analysis by the whole three methods gives a power spectrum with many peaks (Figure 1) and the most on the most appropriate phase curve for the taken method. In some cases, this is not yet the desired rotation period and needs adding analysis in shorter ranges around peaks with a probability high than 40%. The separation of such possible periods is based on two more features. The first phase curve should have two complete peaks and two minimums (Figure 2). The second characteristic is that the shape of the power peak should resemble a Gaussian distribution (Figure 3). These features will be taken into account by analyzing observations obtained by different observatories. Small differences in period values can be combined using a mean-weighted method. Thus, it would be possible to take into account both the different number of observations in the data set and the probability values of the power spectrum peaks obtained in the analysis.

Figure 1

The L–S periodogram for asteroid 4747 Jujo in the range 0.01–40 from Mauna Loa Observatory data.

Figure 2

Calculated light curve for asteroid 4747 Jujo with L–S method from Mauna Loa Observatory data.

Figure 3

The L–S periodogram for asteroid 4747 Jujo.

4.2 5255 Jonsophie

The Main belt asteroid 5255 Jonsophie was discovered at the Palomar observatory by Helin on May 19, 1988. The main obtained parameters of the asteroid in the publications of various authors are very different. The diameter computation of this C- type asteroid varies from 18.14 km detected from the AKARI/IRC Mid-Infrared Asteroid Survey (Usui et al. 2011) catalog to 11.69 km (Massiero et al. 2020) detected from the Near-Earth Object Wide-field Infrared Survey Explorer (NEOWISE) spacecraft data. The values of obtained phase slope G vary from 0.12 to 0.15; absolute magnitude ( H ) from 12.1 to 13.3, and albedo (Al) from 0.03 to 0.09, while Warner in ALCDEF database has given values of G , H , and Al, 0.20, 13.33, and 0.10, respectively, and a smaller diameter of 9.07 km.

167 Baldone observatory observations using G(RP) filter were made in March–May 2022 at phases 17.7–22.3 degrees. The analysis of the 4 h series of the asteroid images with an exposure of 240 s on the nights of March 20, 21, and 22, 2022, and the additional observations of shorter series in April and May with the Fourier row method indicated that the rotation period of the object should be significantly longer than 4 h. The L–S and PDM analysis of data from Catalina Sky Survey, Mauna Loa, Haleakala, and Ponte Uso observatories published in electronic format in MINOR PLANET CIRCULARS/MINOR PLANETS AND COMETS SUPPLEMENT (MPSs) confirmed that the asteroid Jonsophie rotates with a large average period 57.747 ± 0.003  h. Brightness amplitude is close to 2 magnitudes in G(RP) passband (Figures 4 and 5).

Figure 4

Calculated the light curve for asteroid 5255 Jonsophie with L–S method from Mauna Loa Observatory data.

Figure 5

The periodogram for asteroid 5255 Jonsophie.

4.3 (11411) 1999 HK1

The main belt asteroid (11411) 1999 HK1 was discovered during the realization project LINEAR at Lincoln Laboratory’s Experimental Test Site on April 16, 1999. This asteroid has significantly less eccentricity than other main belt objects, and therefore, from time to time, it approaches the Earth closer than the average Earth–Sun distance (closer than 1AU). The closest approach to Earth (Earth MOID) is 0.78600 AU (MPC 2023). The absolute magnitude ( H = 16.21 ) was obtained by (Veres et al. 2015) by analyzing the Pan-STARRS1 telescope 3-year all-sky survey mission data. This previously obtained value has been corrected by Warren (ALCDEF 2022). In the ALCDEF database, he gives the following main parameters for this SE-type asteroid: H = 15.34 , G = 0.30 , Al = 0.3, and a diameter of 2.07 km.

144 Baldone observatory observations using G(RP) filter were made on August 2022 at phases 20.6–21.7 degrees. The analysis of the 4 h continuous monitoring of the asteroid with an exposure of 240 s on the nights of August 11, 12, 13, and 15 in 2022 with the Fourier series method indicated that the rotation period of the object should be significantly longer than 7 h. The data of Mauna Loa observatory L–S and PDM analysis published in electronic format in MINOR PLANET CIRCULARS/MINOR PLANETS AND COMETS SUPPLEMENT (MPSs) brightness give rotation period for asteroid 1999 HK1 9.021 ± 0.002  h with brightness amplitude 0.8 magnitudes in G(RP) passband. Baldone observatory light curve analysis with L–S and PDM methods gives a close period only observed brightness amplitude is less large, 0.9 magnitudes. The amount of published brightness data from Haleakala (74 observ.), and Ponte Uso (79 observ.) observatories and from Catalina Sky Survey (94 observ.) are insufficient for analysis with the PDM method and give inaccurate results. For the same reason, the analysis of the data of the mentioned observatories with the L–S method does not give convincing results (Figures 6 and 7).

Figure 6

Calculated the light curve for asteroid (11411) 1999 HK1 with L–S method from Mauna Loa observatory data.

Figure 7

The periodogram for asteroid (11411) 1999 HK1.

4.4 (15433) 1998 VQ7

The main belt asteroid (15433) 1998 VQ7 was discovered during the realization project LINEAR at Lincoln Laboratory’s Experimental Test Site in December 1998. The absolute magnitude of this V- type asteroid calculated by Veres et al. (2015) is 14.39. Warren 2022 in the ALCDEF database (ALCDEF 2022) gives H = 13.79 which is greater than that obtained by Colazo et al. (2021) from GAIA DR2 release data. Warren in ALCDEF also gives other parameters G = 0.43 , Al = 0.3 , and diameter 3.92 km.

189 Baldone observatory observations using G(RP) filter were made in March and April 2022 at phases 17.7–21.1 degrees. The analysis of the 3–5 h series of the asteroid with an exposure of 240 s on the nights of March 11, 16, 17, and April 01, 02 in 2022 with the F, L–S, and PDM methods gives an average rotation period 3.984 h. The analysis with L–S and PDM methods of the brightness data from The Mauna Loa, Haleakala, and Ponte Uso observatories and Catalina Sky Survey obtained in the time range 2019–2022 and are published in the MINOR PLANET CIRCULARS/MINOR PLANETS AND COMET SUPPLEMENT (MPSs) give a greater mean weighted rotation period 63.787 ± 0.071  h. Brightness amplitude is close to 1.3 magnitudes in G(RP) passband (Figures 8 and 9).

Figure 8

Calculated the light curve for asteroid (15433) 1998 VQ7 with L–S method from Mauna Loa Observatory data.

Figure 9

The periodogram for asteroid (15433) 1998 VQ7.

4.5 (17866) 1998 KV45

The main belt asteroid (17866) 1998 KV45 was discovered during the realization project LINEAR at Lincoln Laboratory’s Experimental Test Site in December 1998. The absolute magnitude of this C- type asteroid calculated by Veres et al. (2015) is 13.68 and by Colazo et al. (2021) is 13.33, while Mainzer et al. (2016) detected from NEOWISE data compilation is 13.00. Mainzer obtained G = 0.15 , l Al = 0.049 , and a diameter of the asteroid 15.15 km. Warren 2022 in ALCDEF gives H = 13.29 , G = 0.12 , Al = 0.057 , and a diameter of 12.23 km.

131 Baldone observatory observations using G(RP) filter were made in March and April 2022 at phases 12.1–12.4 degrees. The analysis of the 3–5 h series of the asteroid with an exposure of 240 s on the nights of March 31 and April 01, 02 in 2022 with the Fourier series, L–S, and PDM methods gives an average rotation period of the object 4.464 h. The analysis with L–S and PDM methods of brightness data from Mauna Loa, Haleakala, Ponte Uso observatories and from Catalina Sky Survey published in SPSs give a slightly smaller rotation period. The weighted period is 4.575 ± 0.005  h. Brightness amplitude is close to 0.6 magnitudes in G(RP) passband (Figures 10 and 11).

Figure 10

Calculated the light curve for asteroid (17866) 1998 KV45 with L–S method from Baldone Observatory data.

Figure 11

The periodogram for asteroid (17866) 1998 KV45.

5 Conclusion

The Fourier series method gives usable results analyzing long series observation in multiple following nights when the rotation period is not longer than 7–10 h. In cases of small series of observations scattered over a large period of time, with uncertainties in brightness, the L–S and PDM methods work more reliably. All three methods can be safely used if the number of observations greatly exceeds a hundred. It should be noted that the PDM method is particularly sensitive to a small number of observations. If the number of observations is less than a hundred, the PDM method mostly does not give good results. The results of the main belt asteroids 4747 Jujo, 5255 Jonsophie, (11411) 1999 HK1, (15433) 1998 VQ7, and (17866) 1998 KV45 brightness analysis have been reported in the Astronomy section of the 81th Annual Scientific Conference of the University of Latvia and in the 5th Anniversary International Conference of NSP FOTONIKA-LV Quantum sciences, Space sciences and Technologies – Photonics 2023 (Eglitis 2023).