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Study on the Spin Characteristics of the Flora Asteroid Family
CHINESE ASTRONOMY AND ASTROPHYSICS Chinese Astronomy and Astrophysics 43 (2019) 375–389
Study on the Spin Characteristics of the Flora Asteroid Family† WANG Yi-bo
LIU Cheng-zhi
FANG Cun-bo
XU Yan
Changcun Observatory, National Astronomical Observatories, Chinese Academy of Sciences, Changchun 130117
Abstract Asteroid families are the remnants of catastrophic collisions, and their fundamental physical properties provide us the information of their parent bodies and thereafter dynamical evolutions. Especially, the orbit and spin characteristics can reveal the influences of the Yarkovsky effect and the YarkovskyO’Keefe-Radzievskii-Paddack (YORP) effect on the evolution of the asteroid family, respectively. Based on the Asteroid Lightcurve Database (LCDB), the spin rate distribution of the Flora asteroid family is studied, and a tendency that the spin rates of the small Flora family members concentrate primarily in the range of 3–5 d−1 is found. The analysis on the spin states of the Flora family asteroids tells that most of these asteroid family members are in the prograde spinning state. However, for the Flora family members with an orbital semi-major axis smaller than 2.2 au, the ratio between the number of prograde spinning members and that of retrograde ones is close to that of the near-Earth asteroids, namely 1 : 3. Furthermore, for those prograde spinning Flora family asteroids with an orbital semi-major axis larger than 2.2 au, a portion of them exhibit the aggregation in the distribution of orbital semi-major axis against the absolute magnitude, and in which nine members show the features similar to the Slivan state. Key words asteroids: general—Flora asteroid family—spin characteristics— Yarkovsky effect—YORP effect 1.
INTRODUCTION
As the “remnants” of the primordial planetesimals that form terrestrial planets and the cores of gaseous giant planets, the asteroids retain a great deal of information about the materials Received 2018–06–26; revised version 2018–09–30
A translation of Acta Astronomica Sinica Vol. 60, No. 1, pp. 1.1–1.12, 2019 [email protected]
0275-1062/19/$-see front matter © 2019 B. V. AllScience rights reserved. c Elsevier 0275-1062/01/$-see front matter 2019 Elsevier B. V. All rights reserved. doi:10.1016/j.chinastron.2018.09.013 PII:
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and environments in the early stage of the solar system. Thus the study on the physical properties of asteroids is of great value in our understanding of planetary formation[1] . The asteroid family, as the final outcome of catastrophic collision, may provide the reliable scenarios for the study of the evolutions of small celestial bodies in the solar system. During the catastrophic collision, the parent body of the asteroid family was fragmented into millions of fragments, which constitute the members of today’s asteroid family. At the same time, because these members are derived from the same parent body, their orbital elements (semimajor axis, eccentricity, and inclination) follow the same distribution. Therefore, the orbital elements of candidate asteroid family members are one of the most important criteria to determine whether they are the members of the family. Thanks to the quick development in astronomy, now our understanding on the small celestial bodies in the solar system has been significantly improved as compared to the last century or even one decade before, however there are still quite many unsolved questions about the formation and evolution mechanisms of an asteroid family. The Yarkovsky effect and the YORP (Yarkovsky-O’Keefe-Radzievskii-Paddack) effect are thought to play a key role in the evolution of small-sized asteroids (generally indicating the small celestial bodies with a diameter smaller than 30–40 km)[2−3] . The Yarkovsky effect arises when a small celestial body with thermal inertia releases the energy absorbed from the sunshine by means of anisotropic thermal radiation, which affects its orbital elements, especially the semi-major axis. The YORP effect, a secular effect mainly caused by the light scattering and heat re-radiation, modifies the rotation of a small celestial body. According to the classical model, the Yarkovsky effect comprises the diurnal effect and the seasonal effect. The diurnal effect changes the orbital variable in positive or negative way, depending on the spin state of the asteroid, while the seasonal effect always leads to the asteroid’s inward migration toward the Sun. Both the Yarkovsky effect and YORP effect are the hottest issues in the current research field of the small celestial bodies in the solar system. The Flora family is the asteroid family proposed by Hirayama in the last century, following the Koronis, Eos and Themis asteroid families[4−5] . The Flora family, being located near the inner edge of the main asteroid belt, is one of the largest asteroid families in the inner main belt[6] . In addition, since the Flora family is very close to the ν6 secular resonance, it may be one of the major contributors to the near-earth asteroids (NEAs) and meteorites, as the research suggested[7] . Meanwhile, the existing studies suggest that the pure collision mechanism cannot produce the ejection velocity high enough so that the family members are driven to cross over the ν6 secular resonance region directly and to reach the distribution region of NEAs. In fact, even the velocity required for reaching the ν6 secular resonance region cannot be achieved only by collision. If the Flora asteroid family is indeed one of the major sources of NEAs, there must be some physical mechanisms other than the collision that can steadily transport the asteroids to the ν6 secular resonance region, so that the Flora family members can be further transferred to the NEA orbit region under the effect of ν6
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secular resonance, in which the Yarkovsky effect plays an important role[8] . Previous studies have clearly shown that the Yarkovsky effect and the YORP effect play an important role in the orbital and rotational evolutions of asteroids, respectively[2−3] . However, our knowledge on their real operating mechanisms is still quite limited. The existing studies only confirm that the Yarkovsky effect and the YORP effect depend on mainly the basic physical properties of asteroids, such as the thermal character, geometrical shape and rotation state, but the specific relationships among them are not very clear yet. Therefore, more observations are needed to provide reliable constraints for the theoretical models. The basic physical properties of the asteroid family are the most important clues to understand the specific actions of the Yarkovsky effect and the YORP effect. In particular, the Yarkovsky effect causes the difference between the distributions of the asteroids with the prograde and retrograde rotations in the same asteroid family; while the YORP effect results in the strong correlation between the orientation of spin axis (thus the obliquity) and the rotation period for the asteroid family members, i.e. the so-called Slivan state, as well as the variation of rotation rate of a small-sized asteroid. All this becomes the indispensable basis for constructing the theoretical models of the Yarkovsky effect and YORP effect[2−3,9] . At present, the research on the physical properties of the Flora family members has attracted wide attention. On the one hand, for the Flora family members, the orbital semimajor axis a has a significant dispersion, which is regarded as the strong evidence of the influence of the Yarkovsky effect on the long-term orbital evolution of asteroids. On the other hand, in the study on the rotation states of the Flora family members, the result given by Kryszczy´ nsky indicated that there is an apparent correlation between the obliquity of spin axis and the rotation period, implying that the Flora asteroid family is in the Slivan state under the influence of the YORP effect[10] . However, the later study made by Hanu´s et al. obtained a conclusion different from that of Kryszczy´ nsky, namely the Flora family [11] members are not in the Slivan state . Obviously, in order to determine whether the Flora family members are in the Slivan state, a larger sample and further systematic analysis are needed. Based on the Asteroid Lightcurve Database (LCDB)[12] , we will study the rotation states of the Flora family members in this paper. With the obtained distribution of spin rates of the Flora family members, and using the presently largest sample of orientations of spin axes of the Flora family members, we mainly investigate the distribution of the obliquities of spin axes and discuss whether these family members are in the Slivan state. This paper is organized as follows. In Section 2, we introduce the LCDB and explain how to select the sample members in the Flora asteroid family in this study. The distribution of spin rates of the family members is analyzed in Section 3, while the distribution of obliquities is discussed in Section 4 and the characteristics of spin states of the family members are analyzed as well. Finally, the main results are summarized in Section 5.
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ASTEROID LIGHTCURVE DATABASE AND THE SELECTION OF THE FLORA FAMILY SAMPLE
Having the most abundant data of the basic physical properties of asteroids, the LCDB is one of the most frequently used databases available currently in the field of asteroid research[12] . It was created most early by Alan W. Harris, and aimed to collect the existing research results about the properties of asteroids (such as the rotation period, geometric albedo, absolute magnitude, category, family attribution, amplitude of light variation, etc.), so as to provide researchers with a reliable statistical sample of asteroids. By March 2018, more than 19,000 records of small celestial bodies in the solar system (including some unnumbered objects) have been included in the LCDB. The Flora asteroid family members defined by the LCDB are distributed in the semimajor axis interval of a ∈ [2.15, 2.35] au (as shown in Fig. 1), a little narrower than the values adopted by the other studies. Considered that other asteroid families may exist around the region where the Flora family resides, in the condition that a sufficient number of sample members can be guaranteed, to appropriately limit the semi-major axis range of the Flora family can reduce the probability that the members of other asteroid families being mixed into the Flora family.
17.5
Absolute Magnitude
15.0 12.5
10.0 C=0.00010 C=0.00020 C=0.00025 C=0.00030 C=0.00040
7.5
2.15
2.20
2.25 Semi-major Axis /au
2.30
2.35
Fig. 1 Distribution of the absolute magnitudes of the Flora family members versus the semi-major axis. The triangles and squares denote the retrograde and prograde rotators, respectively. The five-pointed star indicates the position of the asteroid (8) Flora. The different values of the parameter C for the relationship between the semi-major axis and the absolute magnitude imply the different evolution processes.
By March 2018, more than 2,600 members of the Flora family are listed in the LCDB. In order to ensure the reliability of the study, we select only the sample members with a data quality evaluation parameter of U 2, and we also exclude the asteroids which have
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or potentially have an ambiguous period. For the definitions of the parameter U and the ambiguous period, please refer to Ref. [12] or the description of the LCDB. 3.
ANALYSIS ON THE SPIN RATE
As the most fundamental observable quantity for the study of small celestial bodies in the solar system, the spin rate (rotation period) reflects the current rotational state of an asteroid after a long-term evolution under the actions of the collision and other non-gravitational mechanisms since its formation. Therefore, the statistics on the spin rates of asteroids may improve our understanding of their evolutions and the mechanisms behind. The early studies mainly focused on checking whether the spin rates of the main-belt asteroids can be fitted by the Maxwell distribution. The results show that the spin rates of the asteroids with a large diameter (D > 50 km) can be well approximated by the Maxwell distribution, but for small asteroids, their spin rates deviate significantly from the Maxwell distribution, the ratio of fast rotators to slow rotators being higher than the corresponding value of the Maxwell distribution[13−15] . As the product of catastrophic collision of the parent body, the asteroid family members carried the information about the collision and the succeeding evolution. The study of their spin rates is very important for understanding both the collision and the non-gravitational effects involved thereafter. Binzel et al. summarized the results of the previous studies on the distributions of spin rates of the Eos family and Koronis family[14] , but limited by the knowledge of the evolution of asteroid family at that time, they considered only the effect of the collision process itself in their research. However, the current studies indicated that in addition to the collision, other mechanisms may have significant influence on the evolution of an asteroid family. In particular, the YORP effect caused by the solar irradiation is considered to play a key role in the evolution of the rotational characteristics of an asteroid family[2−3] . From the LCDB, we have selected a sample of the Flora family members with the largest number that can be obtained at present, and by a systematic study on the distribution of spin rates of the Flora family members, confirmed the results given by Kryszczy´ nsky et al[16] . We present in Fig. 2 the distribution (the histogram) of the spin rates of the selected sample by this paper, and the fitted Maxwell distribution (dotted line in Fig. 2) as well. It can be seen from Fig. 2 that the proportions of the Flora family members in the fast spin regime and in the slow spin regime are significantly higher than the corresponding fitting results of the Maxwell distribution, this is consistent with the distribution of spin rates of the main belt asteroids, as the early studies suggested. Meanwhile, it is noteworthy that the proportion of the family members with the spin rates in the range of 5–7 d−1 is apparently lower than that of the Maxwell distribution. The Kolmogorov-Smirnov test (KS test) denies with a confidence of 0.95 the hypothesis that the spin rates of the Flora family members follow the Maxwell distribution, which confirms again that the spin rate distribution of the
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Flora family members cannot be fitted by a Maxwell distribution. In addition, one can also notice in Fig. 2 that except for the relatively high proportion of the slow rotators among the Flora family members, the distribution of spin rates is apparently different from the uniform distribution of spin rates of the main-belt smalldiameter asteroids obtained by Pravec et al.[17] , which is mainly manifested by a significant increase in the number of the asteroids with a spin rate in the range of 2–6 d−1 as compared to other regions. Therefore, we have tried to check whether the high proportion of asteroids with a rotation period of 2–6 d−1 in the distribution found above is caused by the fact that our sample has included much more large-diameter asteroids. For this purpose, we divide the Flora family members into 5 sub-groups with different diameters (D 1 km, 1 km< D 3 km, 3 km< D 15 km, 15 km< D 30 km, and D > 30 km), and analyse the distributions of spin rates in the different sub-samples. As shown in Fig. 3, the relatively high proportion of rotation periods of Flora family members in the range of 2–6 d−1 is not caused by the relatively more large-size members in the sample, but a common feature of the Flora family members with the different sizes.
Probability Density Function
0.25
0.20
0.15
0.10
0.05
0.00
0
2
4
6 Spin Rate /d-1
8
10
12
Fig. 2 Histogram of the spin rate distribution for the Flora asteroid family. The dotted line represents the fitting result of the Maxwell distribution.
In the early studies, Pravec et al.[17] analyzed the spin rates of a sample consisted of the main-belt asteroids with a diameter of 3–15 km, and found that the proportion of slowly-rotating asteroids is significantly higher, while the spin rates of small-sized asteroids follow a uniform distribution. Assuming that for the small-sized asteroids, the variation of spin rate due to the YORP effect is independent to the spin rate itself, they inferred that the spin rates of main-belt asteroids with a 3–15 km diameter obey approximately a uniform distribution after a long-term evolution. We find in our calculation that the spin rates in the sub-sample of 3–15 km diameter exhibit generally an approximately uniform distribution, as shown in Fig. 3(A), which is generally consistent with the result obtained by Pravec et al.,
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but there is no significant “slow rotator excess” as obtained by Pravec et al. And Pravec et al. agreed in their paper that the reason for the significant concentration of slow rotators in the spin rate distribution is not fully understood, but they emphasized that the YORP effect will no longer be the dominant factor for the spin rate evolution of an asteroid after the spin rate of the asteroid with a diameter of 3-15 km has decayed to a certain degree due to the YORP effect. And at this time, the slowly-spinning asteroid is very possible to be in the state of rotation around a non-inertial principal axis. When the timescale of such instable state is comparable to the timescale of the YORP effect, a significant concentration of slow rotators may occur[17] . However, we have not found such slow rotator excess in the Flora family members with a diameter of 3–15 km, possibly implying that the spin rates of most of the Flora family members have not yet been decelerated to the unstable rotating state by the YORP effect, which in turn is consistent with the relatively young age of the Flora family. In addition, the analysis on the spin rates of the Flora family asteroids provides a potential limitation on the timescale for the significant influence of the YORP effect on the rotational evolution of an asteroid, indicating that the typical timescale of the YORP effect for the asteroids of 3–15 km diameter is longer than the age of the Flora family. Comparing Fig. 3(A) with Fig. 3(B), one can easily find a very interesting phenomenon, that is, as the diameter decreases, the spin rate distribution shows a significant concentration at 3–5 d−1 for the Flora family members with a diameter less than 15 km. This reflects the fact that the intensity of the YORP effect changes as the asteroid diameter decreases, and the variation of spin rate of a small asteroid caused by the YORP effect begins to depend on its spin rate itself and exhibit a significant correlation, resulting in a selection effect to the variations of spin rate of small asteroids, as well as a significant concentration in the spin rate distribution of small-diameter asteroids. From Fig. 3(B) we can find also an increase of the proportion of smaller Flora family members in the ultra-slow rotation region. This is because that the intensity of the YORP effect becomes more pronounced as the diameter decreases, so that the spin rates of smaller family members may be decelerated more quickly by the YORP effect than those members of 3–15 km. As a result, these small members enter the unstable rotating state earlier, and the “evolution delay” as pointed out by Pravec et al.[17] happens, thus a corresponding increase of proportion of slow rotators in the ultra-slow spin region appears in the distribution. According to the current theory, the influence of the YORP effect is limited for the small celestial bodies with a larger diameter, but it is difficult to strictly define the specific relationship between its intensity and the asteroid diameter. As the fragments of the parent body after a catastrophic collision, the asteroid family members have different sizes, and thus may provide more information about the influence of the YORP effect. From Figs. 3(C)–(D), we note that all the Flora family members with a diameter larger than 30 km apparently concentrate in a small range in the spin rate distribution, indicating that the rotation period of the Flora family’s parent body may be in the range of 8–24 h. As the diameter decreases, the intensity of the YORP
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Probability Density Function
0.35 0.30 (C) 0.25 0.20 0.15 0.10 0.05 0.00 0 2
12
12
Probability Density Function
Probability Density Function
0.35 0.30 (A) 0.25 0.20 0.15 0.10 0.05 0.00 0 2
Probability Density Function
effect increases obviously for the family members of 15 km < D 30 km, leading to the spin rates of the corresponding family members deviated from the rotation characteristics of their parent body, and therefore the interval of the spin rate distribution is broadened. When the diameters of the family members reduce to the range of 3–15 km, the spin rate distribution attains the maximum interval and exhibits an approximate homogeneity. Comparing Fig. 3(C) with Fig. 3(D), we note that the spin rates of large members of D > 30 km are mainly distributed in the range of 2–3 d−1 , with a median of 2.50 d−1 . For those members of 15 km< D 30 km, the spin rates are concentrated in 2–5 d−1 with a median of 3.36 d−1 . When the diameter goes down to 3–15 km, there is a certain concentration at 4–5 d−1 in an approximately even distribution, and the median spin rate appears at 4.50 d−1 . All these features demonstrate that the YORP effect will not only lead to a broadened spin rate distribution of the Flora family members with a diameter greater than 3 km, but also tend to increase their spin rates. Although this is consistent with the current result of direct detection for the influence of the YORP effect on the spin rates of NEAs, it must be notified that the result of direct detection may be affected by the selection effect in observation. Therefore, more evidence is needed to verify the phenomenon that the YORP effect tends to increase the spin rate. Particularly, we believe that more analyses on the spin characteristics of asteroid family members may provide more effective constraints for this dynamical model.
D=3-15 km D=1-3 km
4 6 8 Spin Rate /d-1
10
D=15-30 km D=3-15 km
4 6 8 Spin Rate /d-1
10
0.35 0.30 (B) 0.25 0.20 0.15 0.10 0.05 0.00 0 2
D=1-3 km Dİ1 km
4 6 8 Spin Rate /d-1
10
12
0.6 0.5 (D)
D˚30 km D=15-30 km
0.4 0.3 0.2 0.1 0.0
0
2
4 Spin Rate /d-1
6
Fig. 3 Histograms of the spin rate distributions for the Flora family asteroids.
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383
ANALYSIS ON THE ORIENTATION OF SPIN AXIS
In the early studies on the distribution of the orientations of spin axes of asteroids, Magnusson analyzed 20 main-belt asteroids with a larger diameter, and found that the longitudes of their spin axes have an approximately uniform distribution[18] . Subsequently, Kryszczy´ nsky et al. showed that the spin axes of asteroids tend to be parallel to the normal of the solar system’s ecliptic plane[19] . Later, Marciniak and Michalowski’s analysis revealed that the distribution of the orientations of spin axes of asteroids does not have a significantly low proportion at the relatively low latitudes, but rather than concentrates in a range of βp > 0◦ , where βp is likely to be at the high-latitude region[20] . Hanuˇs et al. divided the asteroids into groups according to their diameters and studied the distribution of the orientations of spin axes[21] . Their results indicated to a certain degree that the results of Kryszczy´ nsky et[19] and Marciniak & Micha´lowski[20] are valid only when some specific conditions as follows are met. For the asteroids larger than 60 km in diameter, the latitude distribution of spin axes is similar to the result given by Marciniak & Micha´lowski. However, the small-diameter asteroids are more likely consistent with the result given by Kryszczy´ nsky et al[19] . Meanwhile, Slivan studied the spin axes of asteroids of the Koronis family, and found that the orientations of spin axes and the rotation periods of those retrograde-spinning members are spread over a wide range, while for those prograde rotators of this family, the spin axes are concentrated in a relatively narrow region of p ∈ [42◦ , 50◦ ] and the rotation periods are distributed in the range of 7.5–9.5 h, namely in the so-called Slivan state[9] . Thereafter, Vokrouhlick´ y et al. investigated this phenomenon and proposed that such state is caused by the spin-orbit resonance[22] , which in turn is a result of the YORP effect on the Koronis members. Then, after analysing the rotation characteristics of the Flora family, Kryszczy´ nsky suggested that [10] the members of the Flora family are also in the Slivan state , but Hanuˇs et al. denied this conclusion later[11] . The facts mentioned above invoked our interests in the rotation characteristics of the Flora family members. In this study, we select a sample of 124 single Flora asteroids from the LCDB, and analyze the parameters that describe the rotation states of these asteroids. From the orbital elements of these asteroids1 , we obtain the obliquities of spin axes of these asteroids, including totally 71 prograde-spinning asteroids and 53 retrograde ones. The ratio of the prograde to retrograde sources is about 1.34, indicating that the family members are more likely to be the prograde rotators. From Fig. 4, we note that the obliquities of those prograde-spinning members tend to be 20◦ –50◦ , with a median of 37◦ ; while the retrograde rotators are more likely to be found in the range of 130◦ –170◦ , with a median of 149◦ . In comparison, the prograde rotators in the sample have a relatively concentrated distribution of obliquities, which is very similar to the obliquity distribution of spin axes in the Slivan state. Taking the orbital semi-major axis of 2.2 au as the boundary, we divide these member 1 https://minorplanetcenter.net/iau/MPCORB.html
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asteroids into two groups, and analyze separately the prograde and retrograde rotations in the family members on the both sides of the boundary. We find that in the sample there are 27 members in the left side with the semi-major axes smaller than 2.2 au, while there are 97 members, including the asteroid (8) Flora, in the right side with the semi-major axes larger than 2.2 au. Among those members in the left side, the majority tends to have a retrograde rotation, and the proportion occupied by the prograde rotators is as small as 0.296, very close to the 1:3 ratio of the prograde to retrograde rotators in the NEAs[2−3] , implying a potential relationship between the Flora family and the NEAs. Combining with the fact that the Flora family resides closely to the ν6 secular resonance at the inner edge of the main belt, we may suggest that the Flora family is one of the potential sources of the NEAs. 20
Counts
15
10
5
0
0
30
60
90
εp
120
150
180
Fig. 4 Histogram of the obliquity distribution of the orientations of spin axes of the Flora family asteroids
By comparing the obliquity distributions of the Flora family members in Fig. 5, we note that those members with the semi-major axes larger than 2.2 au are more likely to have a prograde rotation, which is consistent with the theoretical prediction of the Yarkovsky effect. Meanwhile, it is noteworthy that for the members with the semi-major axes smaller than 2.2 au, the ratio between the prograde and retrograde rotators is about 2 : 1. More observations and further theoretical studies are required to check this ratio and its implications. Different from the previous studies, we have analyzed separately the distributions of the obliquities of spin axes and the spin rates for the Flora members in the above-mentioned two regions. As shown in Fig. 6(A), for the family members with the semi-major axes smaller than 2.2 au, the prograde rotators are more likely to have a slower rotation than the retrograde rotators. However, taking account of the small number of prograde rotators in this region, the uncertainty of this trend is apparent, and the possibility that it is caused by the selection effect cannot be ruled out. Particularly, as a comparison, in the sample consisted of the family members with the semi-major axes not less than 2.2 au, no similar trend can be found (Fig. 6(B)). In addition, from Fig. 6 we note that although the obliquities of spin axes of the Flora
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family members exhibit a trend of concentration, their rotation periods are distributed in a wide range, indicating that the family members may not be in the Slivan state. Meanwhile, we note that for the prograde rotators in the region of the semi-major axis less than 2.2 au, their orbital semi-major axes are obviously concentrated around 2.2 au (Fig. 1). And relatively, those retrograde rotators have a larger range of semi-major axes.
Probability Density Function
0.020 Semi-major axis˘2.2 au Semi-major axisı2.2 au 0.015
0.010
0.005
0.000
0
30
60
90
120
εp
150
180
Fig. 5 Histogram for the obliquity distributions of the orientations of spin axes of the Flora family asteroids in different semi-major axis ranges.
12
(A)
Spin Rate /d-1
10
0.0004
8 0.0003 6 4
0.0002
0.0004
(B)
10 Spin Rate /d-1
12
8
0.0003
6 0.0002
4 2
2 0.0001 0
0
30
60
90 120 150 180
εp
0 0
0.0001 30
60
90 120 150 180
εp
Fig. 6 Number density distributions in the plane of the obliquity of spin axis versus the spin rate ((A) semi-major axis < 2.2 au and (B) semi-major axis 2.2 au). The gray-value bar indicates the normalized probability density.
However, our further analysis on the distributions of the semi-major axes and absolute magnitudes of these 124 family members reveal that among those family members with a prograde rotation and a semi-major axis larger than 2.2 au, the parameters of some asteroids exhibit an apparent concentration in the distribution. This cannot be explained simply by the selection effect of observation. In addition, we find that these family members exhibit some similar characteristics, namely they are all located in the vicinity of the curve C 0.00010, where C is a parameter in the relationship between the semi-major axis and the absolute magnitude, and their semi-major axes are all crowded in a narrow range. And this phenomenon is usually ignored in the previous studies.
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In order to make a detailed analysis on the family members in the above-mentioned region, we need to set a proper sub-sample. On one hand, this sub-sample should contain a certain number of asteroids to ensure the reliability of our analysis; and on the other hand, we have to minimize the number of possible interlopers. Therefore, only those asteroids satisfying 0.00008 C 0.00013 and a ∈ [2.21, 2.25] au are selected as the sample for further analysis. Finally, a total of 26 family members of prograde rotation are selected, accounting for 0.37 of the total prograde-spinning members. Analyzing the obliquities of spin axes and the rotation periods of these members shows that there are 9 asteroids in this sub-sample, whose obliquities and periods exhibit significant concentrations in distribution, as shown in Fig. 7. The characteristic parameters of these nine Flora asteroids are listed in Table 1. Their obliquities of spin axes are concentrated in 35◦ –40◦ , and their rotation periods are distributed in the the interval of 3–7 h. These characteristics are very similar to the ones of those prograde rotators in the Koronis family, which are found to be in the Slivan state. It seemingly suggests that the strong correlation between the orientation of spin axis and the rotation period is a common phenomenon for all asteroid families influenced by the YORP effect. But for the majority of members of an asteroid family, they must satisfy certain preconditions to have such strong correlation, and these preconditions are most likely related to their original state of rotation. In other words, if the members of an asteroid family were in the Slivan state, their primordial orientations of spin axes and rotation periods must satisfy certain restrictions, which ensure that they will evolve into the state of spin-orbit resonance under the long-term influence of the YORP effect. At the same time, these initial conditions are inevitably related to their semi-major axes, and also should be related to the parameter C in the relationship between the semi-major axis and the absolute magnitude. 19 17 Period /h
15
0.004
13 11
0.003
9
0.002
7
0.001
5
0
3 0
30
εp
60
90
Fig. 7 Number density distribution in the plane of the obliquity of spin axis versus the rotation period. The sizes of the open circles represent the diameters of the given nine asteroids. The gray-value bar indicates the normalized probability density of the distribution.
However, in the selected area for this analysis, only 9 out of the total 26 progradespinning members are in such a state of strong correlation. A possible explanation is that
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for those members not in the state of strong correlation, the initial state of rotation did not fully satisfy the specific conditions, thus they are still in the process of evolution. This also partly explains why this strong correlation in our analysis is related to the parameter C, which depends on the evolution timescale. Meanwhile, this is also consistent with the fact that the Flora asteroid family is a young family[23] . In order to further confirm the above conclusion, we check the spin states and orbit distribution of those prograde-spinning members in the Koronis asteroid family obtained by Slivan[9] , and we find that the semi-major axes and absolute magnitudes of (311) Claudia, (534) Nassovia and (720) Bohlinia are concentrated in a relatively small range, but (1223) Neckar is like an outlier, deviating significantly from the area where the other three reside. We have also noticed that Hanuˇs et al[11] have re-evaluated the orientation of the spin axis of Neckar, and their result has a certain difference from the result given by Silvan. Adopting the new result given by Hanuˇs et al, we estimate again the obliquity of Neckar to be about 61◦ , which is also beyond the range of the Slivan state in the Koronis family. Table 1 Characteristic parameters of the nine Flora family asteroids in the special region (see text) Asteroid
Class
Absolute magnitude
Period
Semi-major axis
Obliquity
/h
/au
/◦
0.2075
4.3160
2.222
37
Albedoa
/mag (291) Alice
S
11.45
(296) Phaetusa
S
12.62
0.24
4.5380
2.229
38
(825) Tanina
S
11.84
0.1508
6.9398
2.226
38
(915) Cosette
S
11.76
0.24
4.4697
2.228
37
(1518) Rovaniemi
S
12.40
0.24
5.2505
2.225
37
(1785) Wurm
S
12.50
0.24
3.2693
2.236
38
(2094) Magnitka
S
12.49
0.1204
6.1122
2.232
37
(3573) Holmberg
S
12.90
0.24
6.5425
2.239
39
(3918) Brel
S
13.03
0.2732
3.0968
2.245
37
a
For those objects whose albedos were not derived from observations, an estimated albedo pv = 0.24 was used in the LCDB.
As a matter of fact, the above results indicate that our current knowledge on the mechanism of the YORP effect is incomplete. Obviously, to be in the Slivan state for the members of an asteroid family requires some specific initial constraints. However, to determine these initial constraints, we need to make the statistical analysis on the spin states of even more asteroid families to provide the more comprehensive evidence. In particular, the relevant studies on the Vesta family, Eos family, and Eunmia family will be able to provide us with more and rich information. Meanwhile, for the Flora family and Koronis family, as the more and more data of the spin states of family members are obtained, the
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size of the sample to be studied will be enlarged, and we will have more exhaustive data for the study of the YORP effect and the Slivan state. 5.
CONCLUSION
The study of spin characteristics of asteroid family members provides us with the reliable evidence for understanding the asteroids’ collision process, the influences of the Yarkovsky effect on the asteroid’s orbital elements, and of the YORP effect on the evolution of asteroid’s spin state. Based on the LCDB, the results of our analysis on the spin characteristics of the Flora family members lead to the conclusions as follows. (1) Based on the sample with a larger size, and by analyzing the distribution of spin rates of the Flora family members, we have further confirmed the results obtained by Kryszczy´ nsky et al that the spin rates of these family members do not follow the Maxwell distribution, and that the numbers of the slowly-spinning members and fast-spinning members in the family far exceed the numbers given by the Maxwell distribution[16] . In addition, we note that in the Flora family, the spin rates of the members with a diameter larger than 15 km are mainly distributed in the range of 1–5 d−1 , while the spin rates of those members with a diameter of 3-15 km have an approximately uniform distribution. As the diameter decreases, the spin rates of those small Flora members are likely to concentrate in the range of 3–5 d−1 . (2) By analyzing the spin axes of the 124 Flora members, we confirm that the rotations of the Flora family members tend to be prograde in the whole. In particular, for those members with the orbital semi-major axes larger than 2.2 au, the majority of them are in a prograde-spinning state, while for the members with the semi-major axes smaller than 2.2 au, inversely they are most likely to be retrograde rotators. This is consistent with the theoretical prediction of the Yarkovsky effect. At the same time, we also note that for the family members with the orbital semi-major axes smaller than 2.2 au, the ratio of retrograde rotators to prograde rotators is close to the ratio that in NEAs, namely 3 : 1, indicating the relation between the Flora family and NEAs, and implying that the former may be one of the major suppliers to the latter. (3) Based on the analysis on the spin states of the Flora family members, we find that not all the family members are in a unified Slivan state. A large number of prograde rotators in the family are aggregated in a very small region on the semi-major axis versus absolute magnitude plane, in which nine members exhibit the characteristics similar to the prograde rotators in the Slivan state. Therefore, we conclude that the Slivan state may exist commonly among many asteroid families. But for a certain asteroid family, it does not mean that all the prograde-spinning members will have a strong correlation between the obliquity of spin axis and the rotation period. Such correlation may be related to the initial spin states of the members just after the formation of the asteroid family. It should be emphasized that only a few members of the family that meet the specific initial constraints may undergo
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the long-term evolution under the YORP effect and eventually attain a relatively stable state of spin-orbit resonance. Particularly, this evolution depends inevitably on the orbital semi-major axis and size of the family member. ACKNOWLEDGEMENTS The authors would like to thank the anonymous referee, whose comments and suggestions have improved this paper very much. References 1
Michel P., DeMeo F. E., Bottke W. F., Asteroids IV, Tucson: University of Arizona Press, 2015, 3-10
2
Vokrouhlick´ y D., Bottke Jr W. F., Rubincam D. P., et al., AREPS, 2006, 34, 157
3
Vokrouhlick´ y D., Bottke W. F., Chesley S. R., et al., The Yarkovsky and YORP Effects//Michel P., DeMeo F. E., Bottke W. F., Asteroids IV, Tucson: University of Arizona Press, 2015, 509-531
4
Hirayama K., AJ, 1918, 31, 185
5
Hirayama K., AOTok, 1919, 1, 52
6
Nesvorn´ y D., Broˇ z M., Carruba V., Identification and Dynamical Properties of Asteroid Families//Michel P., DeMeo F. E., Bottke W. F., Asteroids IV, Tucson: University of Arizona Press, 2015, 297-321
7
Vokrouhlick´ y D., Bottke W. F., Nesvorn´ y D., AJ, 2017, 153, 172
8
Bottke Jr. W. F., Vokrouhlick´ y D., Rubincam D. P., et al., The Effect of Yarkovsky Thermal Forces on the Dynamical Evolution of Asteroids and Meteoroids//Bottke Jr. W. F., Cellino A., Paolicchi P., et al., Asteroids III, Tucson: University of Arizona Press, 2002, 395-408
9
Slivan S. M., Natur, 2002, 419, 49
10
Kryszczy´ nsky A., A&A, 2013, 551, A102
11
ˇ Hanuˇs J., Broˇ z M., Durech J., et al., A&A, 2013, 559, A134
12
Warner B. D., Harris A. W., Pravec P., Icar, 2009, 202, 134
13
Farinella P., Paolicchi P., Zappala V., A&A, 1981, 104, 159
14
Binzel R. P., Farinella P., Zappala V., et al., Asteroid Rotation Rates - Distributions and Statistics//Binzel R. P., Gehrels T., Matthews M. S., Asteroids II, Tucson: University of Arizona Press, 1989, 416-441
15
Pravec P., Harris A. W., Michalowski T., Asteroid Rotations//Bottke Jr. W. F., Cellino A., Paolicchi P., et al., Asteroids III, Tucson: University of Arizona Press, 2002, 113-122
16
Kryszczy´ nsky A., Colas F., Poli´ nska M., et al., A&A, 2012, 546, A72
17
Pravec P., Harris A. W., Vokrouhlick´ y D., et al., Icar, 2008, 197, 497
18
Magnusson P., Icar, 1986, 68, 1
19
Kryszczy´ nsky A., La Spina A., Paolicchi P., et al., Icar, 2007, 192, 223
20
Marciniak A., Micha´lowski T., A&A, 2010, 512, A56
21
ˇ Hanuˇs J., Durech J., Broˇ z M., et al., A&A, 2011, 530, A134
22
Vokrouhlick´ y D., Nesvorn´ y D., Bottke W. F., Natur, 2003, 425, 147
23
Dykhuis M. J., Molnar L., Van Kooten S. J., et al., Icar, 2014, 243, 111
Study on the Spin Characteristics of the Flora Asteroid Family† WANG Yi-bo
LIU Cheng-zhi
FANG Cun-bo
XU Yan
Changcun Observatory, National Astronomical Observatories, Chinese Academy of Sciences, Changchun 130117
Abstract Asteroid families are the remnants of catastrophic collisions, and their fundamental physical properties provide us the information of their parent bodies and thereafter dynamical evolutions. Especially, the orbit and spin characteristics can reveal the influences of the Yarkovsky effect and the YarkovskyO’Keefe-Radzievskii-Paddack (YORP) effect on the evolution of the asteroid family, respectively. Based on the Asteroid Lightcurve Database (LCDB), the spin rate distribution of the Flora asteroid family is studied, and a tendency that the spin rates of the small Flora family members concentrate primarily in the range of 3–5 d−1 is found. The analysis on the spin states of the Flora family asteroids tells that most of these asteroid family members are in the prograde spinning state. However, for the Flora family members with an orbital semi-major axis smaller than 2.2 au, the ratio between the number of prograde spinning members and that of retrograde ones is close to that of the near-Earth asteroids, namely 1 : 3. Furthermore, for those prograde spinning Flora family asteroids with an orbital semi-major axis larger than 2.2 au, a portion of them exhibit the aggregation in the distribution of orbital semi-major axis against the absolute magnitude, and in which nine members show the features similar to the Slivan state. Key words asteroids: general—Flora asteroid family—spin characteristics— Yarkovsky effect—YORP effect 1.
INTRODUCTION
As the “remnants” of the primordial planetesimals that form terrestrial planets and the cores of gaseous giant planets, the asteroids retain a great deal of information about the materials Received 2018–06–26; revised version 2018–09–30
A translation of Acta Astronomica Sinica Vol. 60, No. 1, pp. 1.1–1.12, 2019 [email protected]
0275-1062/19/$-see front matter © 2019 B. V. AllScience rights reserved. c Elsevier 0275-1062/01/$-see front matter 2019 Elsevier B. V. All rights reserved. doi:10.1016/j.chinastron.2018.09.013 PII:
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and environments in the early stage of the solar system. Thus the study on the physical properties of asteroids is of great value in our understanding of planetary formation[1] . The asteroid family, as the final outcome of catastrophic collision, may provide the reliable scenarios for the study of the evolutions of small celestial bodies in the solar system. During the catastrophic collision, the parent body of the asteroid family was fragmented into millions of fragments, which constitute the members of today’s asteroid family. At the same time, because these members are derived from the same parent body, their orbital elements (semimajor axis, eccentricity, and inclination) follow the same distribution. Therefore, the orbital elements of candidate asteroid family members are one of the most important criteria to determine whether they are the members of the family. Thanks to the quick development in astronomy, now our understanding on the small celestial bodies in the solar system has been significantly improved as compared to the last century or even one decade before, however there are still quite many unsolved questions about the formation and evolution mechanisms of an asteroid family. The Yarkovsky effect and the YORP (Yarkovsky-O’Keefe-Radzievskii-Paddack) effect are thought to play a key role in the evolution of small-sized asteroids (generally indicating the small celestial bodies with a diameter smaller than 30–40 km)[2−3] . The Yarkovsky effect arises when a small celestial body with thermal inertia releases the energy absorbed from the sunshine by means of anisotropic thermal radiation, which affects its orbital elements, especially the semi-major axis. The YORP effect, a secular effect mainly caused by the light scattering and heat re-radiation, modifies the rotation of a small celestial body. According to the classical model, the Yarkovsky effect comprises the diurnal effect and the seasonal effect. The diurnal effect changes the orbital variable in positive or negative way, depending on the spin state of the asteroid, while the seasonal effect always leads to the asteroid’s inward migration toward the Sun. Both the Yarkovsky effect and YORP effect are the hottest issues in the current research field of the small celestial bodies in the solar system. The Flora family is the asteroid family proposed by Hirayama in the last century, following the Koronis, Eos and Themis asteroid families[4−5] . The Flora family, being located near the inner edge of the main asteroid belt, is one of the largest asteroid families in the inner main belt[6] . In addition, since the Flora family is very close to the ν6 secular resonance, it may be one of the major contributors to the near-earth asteroids (NEAs) and meteorites, as the research suggested[7] . Meanwhile, the existing studies suggest that the pure collision mechanism cannot produce the ejection velocity high enough so that the family members are driven to cross over the ν6 secular resonance region directly and to reach the distribution region of NEAs. In fact, even the velocity required for reaching the ν6 secular resonance region cannot be achieved only by collision. If the Flora asteroid family is indeed one of the major sources of NEAs, there must be some physical mechanisms other than the collision that can steadily transport the asteroids to the ν6 secular resonance region, so that the Flora family members can be further transferred to the NEA orbit region under the effect of ν6
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secular resonance, in which the Yarkovsky effect plays an important role[8] . Previous studies have clearly shown that the Yarkovsky effect and the YORP effect play an important role in the orbital and rotational evolutions of asteroids, respectively[2−3] . However, our knowledge on their real operating mechanisms is still quite limited. The existing studies only confirm that the Yarkovsky effect and the YORP effect depend on mainly the basic physical properties of asteroids, such as the thermal character, geometrical shape and rotation state, but the specific relationships among them are not very clear yet. Therefore, more observations are needed to provide reliable constraints for the theoretical models. The basic physical properties of the asteroid family are the most important clues to understand the specific actions of the Yarkovsky effect and the YORP effect. In particular, the Yarkovsky effect causes the difference between the distributions of the asteroids with the prograde and retrograde rotations in the same asteroid family; while the YORP effect results in the strong correlation between the orientation of spin axis (thus the obliquity) and the rotation period for the asteroid family members, i.e. the so-called Slivan state, as well as the variation of rotation rate of a small-sized asteroid. All this becomes the indispensable basis for constructing the theoretical models of the Yarkovsky effect and YORP effect[2−3,9] . At present, the research on the physical properties of the Flora family members has attracted wide attention. On the one hand, for the Flora family members, the orbital semimajor axis a has a significant dispersion, which is regarded as the strong evidence of the influence of the Yarkovsky effect on the long-term orbital evolution of asteroids. On the other hand, in the study on the rotation states of the Flora family members, the result given by Kryszczy´ nsky indicated that there is an apparent correlation between the obliquity of spin axis and the rotation period, implying that the Flora asteroid family is in the Slivan state under the influence of the YORP effect[10] . However, the later study made by Hanu´s et al. obtained a conclusion different from that of Kryszczy´ nsky, namely the Flora family [11] members are not in the Slivan state . Obviously, in order to determine whether the Flora family members are in the Slivan state, a larger sample and further systematic analysis are needed. Based on the Asteroid Lightcurve Database (LCDB)[12] , we will study the rotation states of the Flora family members in this paper. With the obtained distribution of spin rates of the Flora family members, and using the presently largest sample of orientations of spin axes of the Flora family members, we mainly investigate the distribution of the obliquities of spin axes and discuss whether these family members are in the Slivan state. This paper is organized as follows. In Section 2, we introduce the LCDB and explain how to select the sample members in the Flora asteroid family in this study. The distribution of spin rates of the family members is analyzed in Section 3, while the distribution of obliquities is discussed in Section 4 and the characteristics of spin states of the family members are analyzed as well. Finally, the main results are summarized in Section 5.
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ASTEROID LIGHTCURVE DATABASE AND THE SELECTION OF THE FLORA FAMILY SAMPLE
Having the most abundant data of the basic physical properties of asteroids, the LCDB is one of the most frequently used databases available currently in the field of asteroid research[12] . It was created most early by Alan W. Harris, and aimed to collect the existing research results about the properties of asteroids (such as the rotation period, geometric albedo, absolute magnitude, category, family attribution, amplitude of light variation, etc.), so as to provide researchers with a reliable statistical sample of asteroids. By March 2018, more than 19,000 records of small celestial bodies in the solar system (including some unnumbered objects) have been included in the LCDB. The Flora asteroid family members defined by the LCDB are distributed in the semimajor axis interval of a ∈ [2.15, 2.35] au (as shown in Fig. 1), a little narrower than the values adopted by the other studies. Considered that other asteroid families may exist around the region where the Flora family resides, in the condition that a sufficient number of sample members can be guaranteed, to appropriately limit the semi-major axis range of the Flora family can reduce the probability that the members of other asteroid families being mixed into the Flora family.
17.5
Absolute Magnitude
15.0 12.5
10.0 C=0.00010 C=0.00020 C=0.00025 C=0.00030 C=0.00040
7.5
2.15
2.20
2.25 Semi-major Axis /au
2.30
2.35
Fig. 1 Distribution of the absolute magnitudes of the Flora family members versus the semi-major axis. The triangles and squares denote the retrograde and prograde rotators, respectively. The five-pointed star indicates the position of the asteroid (8) Flora. The different values of the parameter C for the relationship between the semi-major axis and the absolute magnitude imply the different evolution processes.
By March 2018, more than 2,600 members of the Flora family are listed in the LCDB. In order to ensure the reliability of the study, we select only the sample members with a data quality evaluation parameter of U 2, and we also exclude the asteroids which have
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or potentially have an ambiguous period. For the definitions of the parameter U and the ambiguous period, please refer to Ref. [12] or the description of the LCDB. 3.
ANALYSIS ON THE SPIN RATE
As the most fundamental observable quantity for the study of small celestial bodies in the solar system, the spin rate (rotation period) reflects the current rotational state of an asteroid after a long-term evolution under the actions of the collision and other non-gravitational mechanisms since its formation. Therefore, the statistics on the spin rates of asteroids may improve our understanding of their evolutions and the mechanisms behind. The early studies mainly focused on checking whether the spin rates of the main-belt asteroids can be fitted by the Maxwell distribution. The results show that the spin rates of the asteroids with a large diameter (D > 50 km) can be well approximated by the Maxwell distribution, but for small asteroids, their spin rates deviate significantly from the Maxwell distribution, the ratio of fast rotators to slow rotators being higher than the corresponding value of the Maxwell distribution[13−15] . As the product of catastrophic collision of the parent body, the asteroid family members carried the information about the collision and the succeeding evolution. The study of their spin rates is very important for understanding both the collision and the non-gravitational effects involved thereafter. Binzel et al. summarized the results of the previous studies on the distributions of spin rates of the Eos family and Koronis family[14] , but limited by the knowledge of the evolution of asteroid family at that time, they considered only the effect of the collision process itself in their research. However, the current studies indicated that in addition to the collision, other mechanisms may have significant influence on the evolution of an asteroid family. In particular, the YORP effect caused by the solar irradiation is considered to play a key role in the evolution of the rotational characteristics of an asteroid family[2−3] . From the LCDB, we have selected a sample of the Flora family members with the largest number that can be obtained at present, and by a systematic study on the distribution of spin rates of the Flora family members, confirmed the results given by Kryszczy´ nsky et al[16] . We present in Fig. 2 the distribution (the histogram) of the spin rates of the selected sample by this paper, and the fitted Maxwell distribution (dotted line in Fig. 2) as well. It can be seen from Fig. 2 that the proportions of the Flora family members in the fast spin regime and in the slow spin regime are significantly higher than the corresponding fitting results of the Maxwell distribution, this is consistent with the distribution of spin rates of the main belt asteroids, as the early studies suggested. Meanwhile, it is noteworthy that the proportion of the family members with the spin rates in the range of 5–7 d−1 is apparently lower than that of the Maxwell distribution. The Kolmogorov-Smirnov test (KS test) denies with a confidence of 0.95 the hypothesis that the spin rates of the Flora family members follow the Maxwell distribution, which confirms again that the spin rate distribution of the
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Flora family members cannot be fitted by a Maxwell distribution. In addition, one can also notice in Fig. 2 that except for the relatively high proportion of the slow rotators among the Flora family members, the distribution of spin rates is apparently different from the uniform distribution of spin rates of the main-belt smalldiameter asteroids obtained by Pravec et al.[17] , which is mainly manifested by a significant increase in the number of the asteroids with a spin rate in the range of 2–6 d−1 as compared to other regions. Therefore, we have tried to check whether the high proportion of asteroids with a rotation period of 2–6 d−1 in the distribution found above is caused by the fact that our sample has included much more large-diameter asteroids. For this purpose, we divide the Flora family members into 5 sub-groups with different diameters (D 1 km, 1 km< D 3 km, 3 km< D 15 km, 15 km< D 30 km, and D > 30 km), and analyse the distributions of spin rates in the different sub-samples. As shown in Fig. 3, the relatively high proportion of rotation periods of Flora family members in the range of 2–6 d−1 is not caused by the relatively more large-size members in the sample, but a common feature of the Flora family members with the different sizes.
Probability Density Function
0.25
0.20
0.15
0.10
0.05
0.00
0
2
4
6 Spin Rate /d-1
8
10
12
Fig. 2 Histogram of the spin rate distribution for the Flora asteroid family. The dotted line represents the fitting result of the Maxwell distribution.
In the early studies, Pravec et al.[17] analyzed the spin rates of a sample consisted of the main-belt asteroids with a diameter of 3–15 km, and found that the proportion of slowly-rotating asteroids is significantly higher, while the spin rates of small-sized asteroids follow a uniform distribution. Assuming that for the small-sized asteroids, the variation of spin rate due to the YORP effect is independent to the spin rate itself, they inferred that the spin rates of main-belt asteroids with a 3–15 km diameter obey approximately a uniform distribution after a long-term evolution. We find in our calculation that the spin rates in the sub-sample of 3–15 km diameter exhibit generally an approximately uniform distribution, as shown in Fig. 3(A), which is generally consistent with the result obtained by Pravec et al.,
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but there is no significant “slow rotator excess” as obtained by Pravec et al. And Pravec et al. agreed in their paper that the reason for the significant concentration of slow rotators in the spin rate distribution is not fully understood, but they emphasized that the YORP effect will no longer be the dominant factor for the spin rate evolution of an asteroid after the spin rate of the asteroid with a diameter of 3-15 km has decayed to a certain degree due to the YORP effect. And at this time, the slowly-spinning asteroid is very possible to be in the state of rotation around a non-inertial principal axis. When the timescale of such instable state is comparable to the timescale of the YORP effect, a significant concentration of slow rotators may occur[17] . However, we have not found such slow rotator excess in the Flora family members with a diameter of 3–15 km, possibly implying that the spin rates of most of the Flora family members have not yet been decelerated to the unstable rotating state by the YORP effect, which in turn is consistent with the relatively young age of the Flora family. In addition, the analysis on the spin rates of the Flora family asteroids provides a potential limitation on the timescale for the significant influence of the YORP effect on the rotational evolution of an asteroid, indicating that the typical timescale of the YORP effect for the asteroids of 3–15 km diameter is longer than the age of the Flora family. Comparing Fig. 3(A) with Fig. 3(B), one can easily find a very interesting phenomenon, that is, as the diameter decreases, the spin rate distribution shows a significant concentration at 3–5 d−1 for the Flora family members with a diameter less than 15 km. This reflects the fact that the intensity of the YORP effect changes as the asteroid diameter decreases, and the variation of spin rate of a small asteroid caused by the YORP effect begins to depend on its spin rate itself and exhibit a significant correlation, resulting in a selection effect to the variations of spin rate of small asteroids, as well as a significant concentration in the spin rate distribution of small-diameter asteroids. From Fig. 3(B) we can find also an increase of the proportion of smaller Flora family members in the ultra-slow rotation region. This is because that the intensity of the YORP effect becomes more pronounced as the diameter decreases, so that the spin rates of smaller family members may be decelerated more quickly by the YORP effect than those members of 3–15 km. As a result, these small members enter the unstable rotating state earlier, and the “evolution delay” as pointed out by Pravec et al.[17] happens, thus a corresponding increase of proportion of slow rotators in the ultra-slow spin region appears in the distribution. According to the current theory, the influence of the YORP effect is limited for the small celestial bodies with a larger diameter, but it is difficult to strictly define the specific relationship between its intensity and the asteroid diameter. As the fragments of the parent body after a catastrophic collision, the asteroid family members have different sizes, and thus may provide more information about the influence of the YORP effect. From Figs. 3(C)–(D), we note that all the Flora family members with a diameter larger than 30 km apparently concentrate in a small range in the spin rate distribution, indicating that the rotation period of the Flora family’s parent body may be in the range of 8–24 h. As the diameter decreases, the intensity of the YORP
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Probability Density Function
0.35 0.30 (C) 0.25 0.20 0.15 0.10 0.05 0.00 0 2
12
12
Probability Density Function
Probability Density Function
0.35 0.30 (A) 0.25 0.20 0.15 0.10 0.05 0.00 0 2
Probability Density Function
effect increases obviously for the family members of 15 km < D 30 km, leading to the spin rates of the corresponding family members deviated from the rotation characteristics of their parent body, and therefore the interval of the spin rate distribution is broadened. When the diameters of the family members reduce to the range of 3–15 km, the spin rate distribution attains the maximum interval and exhibits an approximate homogeneity. Comparing Fig. 3(C) with Fig. 3(D), we note that the spin rates of large members of D > 30 km are mainly distributed in the range of 2–3 d−1 , with a median of 2.50 d−1 . For those members of 15 km< D 30 km, the spin rates are concentrated in 2–5 d−1 with a median of 3.36 d−1 . When the diameter goes down to 3–15 km, there is a certain concentration at 4–5 d−1 in an approximately even distribution, and the median spin rate appears at 4.50 d−1 . All these features demonstrate that the YORP effect will not only lead to a broadened spin rate distribution of the Flora family members with a diameter greater than 3 km, but also tend to increase their spin rates. Although this is consistent with the current result of direct detection for the influence of the YORP effect on the spin rates of NEAs, it must be notified that the result of direct detection may be affected by the selection effect in observation. Therefore, more evidence is needed to verify the phenomenon that the YORP effect tends to increase the spin rate. Particularly, we believe that more analyses on the spin characteristics of asteroid family members may provide more effective constraints for this dynamical model.
D=3-15 km D=1-3 km
4 6 8 Spin Rate /d-1
10
D=15-30 km D=3-15 km
4 6 8 Spin Rate /d-1
10
0.35 0.30 (B) 0.25 0.20 0.15 0.10 0.05 0.00 0 2
D=1-3 km Dİ1 km
4 6 8 Spin Rate /d-1
10
12
0.6 0.5 (D)
D˚30 km D=15-30 km
0.4 0.3 0.2 0.1 0.0
0
2
4 Spin Rate /d-1
6
Fig. 3 Histograms of the spin rate distributions for the Flora family asteroids.
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4.
383
ANALYSIS ON THE ORIENTATION OF SPIN AXIS
In the early studies on the distribution of the orientations of spin axes of asteroids, Magnusson analyzed 20 main-belt asteroids with a larger diameter, and found that the longitudes of their spin axes have an approximately uniform distribution[18] . Subsequently, Kryszczy´ nsky et al. showed that the spin axes of asteroids tend to be parallel to the normal of the solar system’s ecliptic plane[19] . Later, Marciniak and Michalowski’s analysis revealed that the distribution of the orientations of spin axes of asteroids does not have a significantly low proportion at the relatively low latitudes, but rather than concentrates in a range of βp > 0◦ , where βp is likely to be at the high-latitude region[20] . Hanuˇs et al. divided the asteroids into groups according to their diameters and studied the distribution of the orientations of spin axes[21] . Their results indicated to a certain degree that the results of Kryszczy´ nsky et[19] and Marciniak & Micha´lowski[20] are valid only when some specific conditions as follows are met. For the asteroids larger than 60 km in diameter, the latitude distribution of spin axes is similar to the result given by Marciniak & Micha´lowski. However, the small-diameter asteroids are more likely consistent with the result given by Kryszczy´ nsky et al[19] . Meanwhile, Slivan studied the spin axes of asteroids of the Koronis family, and found that the orientations of spin axes and the rotation periods of those retrograde-spinning members are spread over a wide range, while for those prograde rotators of this family, the spin axes are concentrated in a relatively narrow region of p ∈ [42◦ , 50◦ ] and the rotation periods are distributed in the range of 7.5–9.5 h, namely in the so-called Slivan state[9] . Thereafter, Vokrouhlick´ y et al. investigated this phenomenon and proposed that such state is caused by the spin-orbit resonance[22] , which in turn is a result of the YORP effect on the Koronis members. Then, after analysing the rotation characteristics of the Flora family, Kryszczy´ nsky suggested that [10] the members of the Flora family are also in the Slivan state , but Hanuˇs et al. denied this conclusion later[11] . The facts mentioned above invoked our interests in the rotation characteristics of the Flora family members. In this study, we select a sample of 124 single Flora asteroids from the LCDB, and analyze the parameters that describe the rotation states of these asteroids. From the orbital elements of these asteroids1 , we obtain the obliquities of spin axes of these asteroids, including totally 71 prograde-spinning asteroids and 53 retrograde ones. The ratio of the prograde to retrograde sources is about 1.34, indicating that the family members are more likely to be the prograde rotators. From Fig. 4, we note that the obliquities of those prograde-spinning members tend to be 20◦ –50◦ , with a median of 37◦ ; while the retrograde rotators are more likely to be found in the range of 130◦ –170◦ , with a median of 149◦ . In comparison, the prograde rotators in the sample have a relatively concentrated distribution of obliquities, which is very similar to the obliquity distribution of spin axes in the Slivan state. Taking the orbital semi-major axis of 2.2 au as the boundary, we divide these member 1 https://minorplanetcenter.net/iau/MPCORB.html
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asteroids into two groups, and analyze separately the prograde and retrograde rotations in the family members on the both sides of the boundary. We find that in the sample there are 27 members in the left side with the semi-major axes smaller than 2.2 au, while there are 97 members, including the asteroid (8) Flora, in the right side with the semi-major axes larger than 2.2 au. Among those members in the left side, the majority tends to have a retrograde rotation, and the proportion occupied by the prograde rotators is as small as 0.296, very close to the 1:3 ratio of the prograde to retrograde rotators in the NEAs[2−3] , implying a potential relationship between the Flora family and the NEAs. Combining with the fact that the Flora family resides closely to the ν6 secular resonance at the inner edge of the main belt, we may suggest that the Flora family is one of the potential sources of the NEAs. 20
Counts
15
10
5
0
0
30
60
90
εp
120
150
180
Fig. 4 Histogram of the obliquity distribution of the orientations of spin axes of the Flora family asteroids
By comparing the obliquity distributions of the Flora family members in Fig. 5, we note that those members with the semi-major axes larger than 2.2 au are more likely to have a prograde rotation, which is consistent with the theoretical prediction of the Yarkovsky effect. Meanwhile, it is noteworthy that for the members with the semi-major axes smaller than 2.2 au, the ratio between the prograde and retrograde rotators is about 2 : 1. More observations and further theoretical studies are required to check this ratio and its implications. Different from the previous studies, we have analyzed separately the distributions of the obliquities of spin axes and the spin rates for the Flora members in the above-mentioned two regions. As shown in Fig. 6(A), for the family members with the semi-major axes smaller than 2.2 au, the prograde rotators are more likely to have a slower rotation than the retrograde rotators. However, taking account of the small number of prograde rotators in this region, the uncertainty of this trend is apparent, and the possibility that it is caused by the selection effect cannot be ruled out. Particularly, as a comparison, in the sample consisted of the family members with the semi-major axes not less than 2.2 au, no similar trend can be found (Fig. 6(B)). In addition, from Fig. 6 we note that although the obliquities of spin axes of the Flora
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family members exhibit a trend of concentration, their rotation periods are distributed in a wide range, indicating that the family members may not be in the Slivan state. Meanwhile, we note that for the prograde rotators in the region of the semi-major axis less than 2.2 au, their orbital semi-major axes are obviously concentrated around 2.2 au (Fig. 1). And relatively, those retrograde rotators have a larger range of semi-major axes.
Probability Density Function
0.020 Semi-major axis˘2.2 au Semi-major axisı2.2 au 0.015
0.010
0.005
0.000
0
30
60
90
120
εp
150
180
Fig. 5 Histogram for the obliquity distributions of the orientations of spin axes of the Flora family asteroids in different semi-major axis ranges.
12
(A)
Spin Rate /d-1
10
0.0004
8 0.0003 6 4
0.0002
0.0004
(B)
10 Spin Rate /d-1
12
8
0.0003
6 0.0002
4 2
2 0.0001 0
0
30
60
90 120 150 180
εp
0 0
0.0001 30
60
90 120 150 180
εp
Fig. 6 Number density distributions in the plane of the obliquity of spin axis versus the spin rate ((A) semi-major axis < 2.2 au and (B) semi-major axis 2.2 au). The gray-value bar indicates the normalized probability density.
However, our further analysis on the distributions of the semi-major axes and absolute magnitudes of these 124 family members reveal that among those family members with a prograde rotation and a semi-major axis larger than 2.2 au, the parameters of some asteroids exhibit an apparent concentration in the distribution. This cannot be explained simply by the selection effect of observation. In addition, we find that these family members exhibit some similar characteristics, namely they are all located in the vicinity of the curve C 0.00010, where C is a parameter in the relationship between the semi-major axis and the absolute magnitude, and their semi-major axes are all crowded in a narrow range. And this phenomenon is usually ignored in the previous studies.
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In order to make a detailed analysis on the family members in the above-mentioned region, we need to set a proper sub-sample. On one hand, this sub-sample should contain a certain number of asteroids to ensure the reliability of our analysis; and on the other hand, we have to minimize the number of possible interlopers. Therefore, only those asteroids satisfying 0.00008 C 0.00013 and a ∈ [2.21, 2.25] au are selected as the sample for further analysis. Finally, a total of 26 family members of prograde rotation are selected, accounting for 0.37 of the total prograde-spinning members. Analyzing the obliquities of spin axes and the rotation periods of these members shows that there are 9 asteroids in this sub-sample, whose obliquities and periods exhibit significant concentrations in distribution, as shown in Fig. 7. The characteristic parameters of these nine Flora asteroids are listed in Table 1. Their obliquities of spin axes are concentrated in 35◦ –40◦ , and their rotation periods are distributed in the the interval of 3–7 h. These characteristics are very similar to the ones of those prograde rotators in the Koronis family, which are found to be in the Slivan state. It seemingly suggests that the strong correlation between the orientation of spin axis and the rotation period is a common phenomenon for all asteroid families influenced by the YORP effect. But for the majority of members of an asteroid family, they must satisfy certain preconditions to have such strong correlation, and these preconditions are most likely related to their original state of rotation. In other words, if the members of an asteroid family were in the Slivan state, their primordial orientations of spin axes and rotation periods must satisfy certain restrictions, which ensure that they will evolve into the state of spin-orbit resonance under the long-term influence of the YORP effect. At the same time, these initial conditions are inevitably related to their semi-major axes, and also should be related to the parameter C in the relationship between the semi-major axis and the absolute magnitude. 19 17 Period /h
15
0.004
13 11
0.003
9
0.002
7
0.001
5
0
3 0
30
εp
60
90
Fig. 7 Number density distribution in the plane of the obliquity of spin axis versus the rotation period. The sizes of the open circles represent the diameters of the given nine asteroids. The gray-value bar indicates the normalized probability density of the distribution.
However, in the selected area for this analysis, only 9 out of the total 26 progradespinning members are in such a state of strong correlation. A possible explanation is that
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for those members not in the state of strong correlation, the initial state of rotation did not fully satisfy the specific conditions, thus they are still in the process of evolution. This also partly explains why this strong correlation in our analysis is related to the parameter C, which depends on the evolution timescale. Meanwhile, this is also consistent with the fact that the Flora asteroid family is a young family[23] . In order to further confirm the above conclusion, we check the spin states and orbit distribution of those prograde-spinning members in the Koronis asteroid family obtained by Slivan[9] , and we find that the semi-major axes and absolute magnitudes of (311) Claudia, (534) Nassovia and (720) Bohlinia are concentrated in a relatively small range, but (1223) Neckar is like an outlier, deviating significantly from the area where the other three reside. We have also noticed that Hanuˇs et al[11] have re-evaluated the orientation of the spin axis of Neckar, and their result has a certain difference from the result given by Silvan. Adopting the new result given by Hanuˇs et al, we estimate again the obliquity of Neckar to be about 61◦ , which is also beyond the range of the Slivan state in the Koronis family. Table 1 Characteristic parameters of the nine Flora family asteroids in the special region (see text) Asteroid
Class
Absolute magnitude
Period
Semi-major axis
Obliquity
/h
/au
/◦
0.2075
4.3160
2.222
37
Albedoa
/mag (291) Alice
S
11.45
(296) Phaetusa
S
12.62
0.24
4.5380
2.229
38
(825) Tanina
S
11.84
0.1508
6.9398
2.226
38
(915) Cosette
S
11.76
0.24
4.4697
2.228
37
(1518) Rovaniemi
S
12.40
0.24
5.2505
2.225
37
(1785) Wurm
S
12.50
0.24
3.2693
2.236
38
(2094) Magnitka
S
12.49
0.1204
6.1122
2.232
37
(3573) Holmberg
S
12.90
0.24
6.5425
2.239
39
(3918) Brel
S
13.03
0.2732
3.0968
2.245
37
a
For those objects whose albedos were not derived from observations, an estimated albedo pv = 0.24 was used in the LCDB.
As a matter of fact, the above results indicate that our current knowledge on the mechanism of the YORP effect is incomplete. Obviously, to be in the Slivan state for the members of an asteroid family requires some specific initial constraints. However, to determine these initial constraints, we need to make the statistical analysis on the spin states of even more asteroid families to provide the more comprehensive evidence. In particular, the relevant studies on the Vesta family, Eos family, and Eunmia family will be able to provide us with more and rich information. Meanwhile, for the Flora family and Koronis family, as the more and more data of the spin states of family members are obtained, the
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size of the sample to be studied will be enlarged, and we will have more exhaustive data for the study of the YORP effect and the Slivan state. 5.
CONCLUSION
The study of spin characteristics of asteroid family members provides us with the reliable evidence for understanding the asteroids’ collision process, the influences of the Yarkovsky effect on the asteroid’s orbital elements, and of the YORP effect on the evolution of asteroid’s spin state. Based on the LCDB, the results of our analysis on the spin characteristics of the Flora family members lead to the conclusions as follows. (1) Based on the sample with a larger size, and by analyzing the distribution of spin rates of the Flora family members, we have further confirmed the results obtained by Kryszczy´ nsky et al that the spin rates of these family members do not follow the Maxwell distribution, and that the numbers of the slowly-spinning members and fast-spinning members in the family far exceed the numbers given by the Maxwell distribution[16] . In addition, we note that in the Flora family, the spin rates of the members with a diameter larger than 15 km are mainly distributed in the range of 1–5 d−1 , while the spin rates of those members with a diameter of 3-15 km have an approximately uniform distribution. As the diameter decreases, the spin rates of those small Flora members are likely to concentrate in the range of 3–5 d−1 . (2) By analyzing the spin axes of the 124 Flora members, we confirm that the rotations of the Flora family members tend to be prograde in the whole. In particular, for those members with the orbital semi-major axes larger than 2.2 au, the majority of them are in a prograde-spinning state, while for the members with the semi-major axes smaller than 2.2 au, inversely they are most likely to be retrograde rotators. This is consistent with the theoretical prediction of the Yarkovsky effect. At the same time, we also note that for the family members with the orbital semi-major axes smaller than 2.2 au, the ratio of retrograde rotators to prograde rotators is close to the ratio that in NEAs, namely 3 : 1, indicating the relation between the Flora family and NEAs, and implying that the former may be one of the major suppliers to the latter. (3) Based on the analysis on the spin states of the Flora family members, we find that not all the family members are in a unified Slivan state. A large number of prograde rotators in the family are aggregated in a very small region on the semi-major axis versus absolute magnitude plane, in which nine members exhibit the characteristics similar to the prograde rotators in the Slivan state. Therefore, we conclude that the Slivan state may exist commonly among many asteroid families. But for a certain asteroid family, it does not mean that all the prograde-spinning members will have a strong correlation between the obliquity of spin axis and the rotation period. Such correlation may be related to the initial spin states of the members just after the formation of the asteroid family. It should be emphasized that only a few members of the family that meet the specific initial constraints may undergo
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the long-term evolution under the YORP effect and eventually attain a relatively stable state of spin-orbit resonance. Particularly, this evolution depends inevitably on the orbital semi-major axis and size of the family member. ACKNOWLEDGEMENTS The authors would like to thank the anonymous referee, whose comments and suggestions have improved this paper very much. References 1
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