Asteroid family initial velocity V-shapes

Initial velocity V-shapes of young asteroid families

Bryce T. Bolin11affiliation: Laboratoire Lagrange, Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Blvd. de l’Observatoire, CS 34229, 06304 Nice cedex 4, France 22affiliation: B612 Asteroid Institute, 20 Sunnyside Ave, Suite 427, Mill Valley, CA, 94941, United States 111bryce.bolin@oca.eu, Kevin J. Walsh33affiliation: Southwest Research Institute, 1050 Walnut St. Suite 300, Boulder, CO 80302, United States , Alessandro Morbidelli11affiliation: Laboratoire Lagrange, Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Blvd. de l’Observatoire, CS 34229, 06304 Nice cedex 4, France , Marco Delbó11affiliation: Laboratoire Lagrange, Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Blvd. de l’Observatoire, CS 34229, 06304 Nice cedex 4, France
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Abstract

Ejection velocity fields of asteroid families are largely unconstrained due to the fact that members disperse relatively quickly on Myr time-scales by secular resonances and the Yarkovsky effect. The spreading of fragments in by the Yarkovsky effect is indistinguishable from the spreading caused by the initial ejection of fragments. By examining families 20 Myrs-old, we can use the V-shape identification technique to separate family shapes that are due to the initial ejection velocity field and those that are due to the Yarkovsky effect. 20 Myr-old asteroid families provide an opportunity to study the velocity field of family fragments before they become too dispersed. Only the Karin family’s initial velocity field has been determined and scales inversely with diameter, . We have applied the V-shape identification technique to constrain young families’ initial ejection velocity fields by measuring the curvature of their fragments’ V-shape correlation in semi-major axis, , vs. space. Curvature from a straight line implies a deviation from a scaling of . We measure the V-shape curvature of 11 young asteroid families including the 1993 FY, Aeolia, Brangane, Brasilia, Clarissa, Iannini, Karin, Konig, Koronis(2), Theobalda and Veritas asteroid families. We find that the majority of asteroid families have initial ejection velocity fields consistent with supporting laboratory impact experiments and computer simulations of disrupting asteroid parent bodies.

slugcomment: 68 Pages, 25 Figures, 2 Tables

Key words: minor planets, asteroids, general

Proposed Running Head: Initial velocity V-shapes of young Main Belt asteroid families

Editorial correspondence to:
Bryce Bolin
Observatoire de la Cote d’Azur
Boulevard de l’Observatoire
CS 34229
06304 Nice, France
Phone: +33 04 92 00 30 81
Fax: +33 (0) 4 92 00 30 33
E-mail: bbolin@oca.eu

1 Introduction

Asteroid families are formed as a result of collisional disruptions and cratering events on larger parent bodies (e.g. Durda et al., 2004; Michel et al., 2015). Although dispersed in space, the family members typically cluster in their proper orbital elements, semi-major axis , eccentricity and inclination , close to that of the parent body (e.g. Hirayama, 1918; Zappalà et al., ; Nesvorný et al., 2015) and share similar spectral and reflectance properties (Cellino et al., 2002; Masiero et al., 2013; de León et al., 2016).

It was thought that asteroid fragments remained stationary in orbital elements space after their disruption (Zappalà et al., 1996; Cellino et al., 1999) requiring large ejection velocities to explain the wide dispersion of asteroid family fragments in orbital elements space. Impact simulations and asteroids observed on temporary, unstable orbits as well as family fragments leaking through mean motion and secular resonances provided evidence that asteroids’ orbits were modified due to recoil from anisotropic surface emission of thermal photons, i.e., the Yarkovsky effect was responsible for the large dispersion of asteroids in orbital elements space (Michel et al., 2001; Bottke et al., 2001). There is a degeneracy between the contribution of the initial spreading of an asteroid family fragments’ orbital elements caused by the initial ejection of fragments and the contribution caused by the subsequent drift in caused by the Yarkovsky effect. This degeneracy can only be broken in special cases, such as asteroids leaking through resonances or families that are too disperse to show the imprint of the initial ejection of fragments.

In addition to the cases above, young asteroid families can provide an opportunity to determine how ejection velocities of asteroid family fragments are distributed as a function of their size. The ejection velocities of Karin and Koronis asteroid family fragments have been measured to scale inversely with by the use of Gauss’ equations (Zappalà et al., 1990) or by linking the distribution of family fragments to their out of plane velocities Nesvorný et al. (2002); Carruba et al. (2016a). In this paper we study the least dispersed families in the asteroid belt, which most likely are the youngest, and examine how the spreading of their members depends on size. In an accompanying paper (Bolin et al., 2017) we show that semi-major axis spreading of the oldest and most disperse families follows a different dependence with the fragments’ size. Thus, we demonstrate that the shape of a family in the vs. space is a generic way to break the degeneracy between initial ejection velocity and Yarkovsky evolution and can be used to tell the relative contribution of each of these processes.

2 Initial velocity field V-shapes

The displacement in after the disruption of a parent body, where is the location in of the parent body, is a function of , the transverse velocity component of the ejected fragment and its parent body’s mean motion, (Zappalà et al., 1996)

(1)

The initial ejection of the family fragments should result in a symmetric V-shaped spread of fragments in vs. the reciprocal of the diameter, , space because scales inversely with asteroid diameter (Cellino et al., 1999; Vokrouhlický et al., 2006a)

(2)

is equal to 1329 km where is a parameter that describes the width of the fragment ejection velocity distribution (Michel et al., 2004; Nesvorný et al., 2006; Vokrouhlický et al., 2006a, b; Durda et al., 2007). for known asteroid families such as Karin and Erigone range between 15 and 50 (Nesvorný et al., 2006; Bottke et al., 2007; Carruba and Morbidelli, 2011; Masiero et al., 2012; Nesvorný et al., 2015).

is determined by modedlling the initial ejection of fragments according to Eq.2 where is the exponent scaling with (Vokrouhlický et al., 2006a, b). = 1 would imply a simple dependence. Modedlling of the ejection of fragments is done for asteroid families younger than 20 Myrs where the Yarkovsky effect has not had enough time to modify the of the fragments such as for the Karin asteroid family (Nesvorný et al., 2006). The modedlling of the initial ejection velocities of the fragments includes evolution of family fragments’ according to the Yarkovsky and YORP effects (Bottke et al., 2006; Vokrouhlický et al., 2015). Additional constraints to the initial ejection velocity field can be provided by the distribution of an asteroid family, which is supposed to remain essentially unaltered during the Yarkosky evolution (Carruba and Nesvorný, 2016; Carruba et al., 2016a). For cases where can not be determined, the escape velocity of the asteroid family parent body is used for (Vokrouhlický et al., 2006a; Walsh et al., 2013). The escape velocity of the parent body is a good estimate for because most particles are ejected from their parent bodies at velocities around the escape velocity of the parent body in numerical impact simulations (Durda et al., 2007; Ševeček et al., 2017).

Karin and Koronis asteroid family data suggest that (Nesvorný et al., 2002, 2006; Carruba et al., 2016a) while analytical calculations of can be as high as 1.5 (Cellino et al., 1999). is the angle of the fragment’s velocity relative the transverse direction of the parent body’s orbit. In Eq. 2, is expected to be uniformly distributed between -1 and 1. We assume that the number of fragments is high enough that the V-shape’s edge is defined by fragments with or -1.0 in Eq. 2. The of fragments interior to the V-shape border with the same value of 1 will scale with by the same as fragments on the V-shape border. The left side of Eq. 2 is decreased for V-shapes with fewer fragments by a factor of because is the average value of between the intervals 0 and and and resulting in a distorted value of .

We re-write Eq. 2 in vs. space with to obtain as a function of and

(3)

The spread in family fragments by their initial ejection from the parent body has the same functional form of the spreading of family fragments caused by the Yarkovksy effect (Bottke et al., 2006; Vokrouhlický et al., 2015).

(4)

is the width of the V-shape for a specific value of . is normalised to the width of the V-shape with

(5)

Eq. 4 is re-written in terms of by using Eq. 5

(6)

The parameters, and in Eqs. 2 and 6, describing the shape of family V-shapes in vs. caused by the dependent initial ejection of fragments or Yakovsky force are functionally equivalent.

We combine Eqs. 3 and 6 to obtain the spread of the V-shape, C, caused by the initial ejection of fragments as a function of and

(7)

We apply the techniques to identify asteroid family Yarkovsky V-shapes as defined in Bolin et al. (2017) to measure the of an asteroid family’s initial ejection velocity V-shape because the shape of the initial ejection velocity field as described by Eq. 3 is similar to shape acquired by the Yarkovsky effect described by Eq. 6.

2.1 V-shape identification technique and measurement of

The identification of the family V-shape is performed by determining , and for a family V-shape according to Eq. 6 in vs. space using a modified version of the border method from Bolin et al. (2017).

(8)
(9)

Eqs. 8 and 9 are normalised the area in vs. between the nominal and outer V-shapes defined by and in the denominator for Eq. 8 and between the nominal and inner V-shapes defined by defined by and in the denominator for Eq. 9.

The symbol in Eqs.8 and 9 indicates summation on the asteroids of the catalogue, with semi-major axis and reciprocal diameter . The symbol indicates Dirac’s function, and and are the low and high semi-major axis range in which the asteroid catalogue is considered. The function weighs the right-side portions of Eqs. 8 and 9 by their size so that the location of the V-shape in vs. space will be weighted towards its larger members. The exponent 2.5 is used for , in agreement with the cumulative size distribution of collisionally relaxed populations and with the observed distribution for MBAs in the range (Jedicke et al., 2002).

Walsh et al. (2013) found that the borders of the V-shapes of the Eulalia and new Polana family could be identified by the peak in the ratio where and are the number of asteroids falling between the curves defined by Eq. 6 assuming for values and and and , respectively, with and . We extend our technique to search for a peak in the ratio , which corresponds to weighting the ratio of by the value of . This approach has been shown to provide sharper results (Delbo et al., 2017). Here we extend the search for a maximum of to 3 dimensions, in the , and space. A peak value in as seen in the top panel of Fig. 1 for the synthetic astroid family described in Section 3.1. For simplicity only the projection on the , plane compared to Bolin et al. (2017) and Delbo et al. (2017) that used only the projection in the , plane) indicates the best-fitting values of , and for a family V-shape using Eq. 6 (bottom panel of Fig. 1).

Figure 1: Application of the V-shape identification to synthetic asteroid family data at Time = 0. (Top panel) The ratio of to ratio in the - range, (,) where is equal to au and , not to be confused with , is equal to au, for the single synthetic family. The box marks the peak value in for the synthetic family V-shape. (Bottom Panel) is plotted for the peak value with the primary V-shape as a solid line where . The dashed lines mark the boundaries for the area in vs. space for and using Eq. 6, where = 2.366 au and au.

The value of is used similarly as in Bolin et al. (2017). The value of dC used depends on the density of asteroids on the family V-shape edge. The value of dC can be a few 10 of the V-shape’s value if the density of asteroids on the V-shape edge is high such as the case of the Karin family (see the bottom panel of Fig. 2) and more, up to 4050 if the V-shape edge is more diffuse such as in the case of the Brangane family (see the bottom panel of Figs. 3) (Milani et al., 2014; Nesvorný et al., 2015; Spoto et al., 2015). The inner and outer V-shapes must be wide enough to include enough asteroids in the inner V-shape and measure a to ratio high enough to identify the family V-shape. The V-shape can include interlopers or asteroids which are not part of the family V-shape if the value used for is used is too large (Nesvorný et al., 2015; Radović et al., 2017). The V-shape identification technique was tested on families identified by both Nesvorný et al. (2015) and Milani et al. (2014) to verify that the V-shape , and determination works on family membership definitions from either database and produces similar results. The V-shape , and determination technique was tested the 1993 FY, Aeolia, Brangane, Brasilia, Iannini and König families identified by both Nesvorný et al. (2015) and Milani et al. (2014) discussed in Sections A.1, A.2, A.3, A.4, A.6 and A.8.

Figure 2: The same as Fig. 1 for Karin asteroid family data from Nesvorný et al. (2015). (Top panel) is equal to au and , is equal to au. (Bottom Panel) is plotted with , = 2.865 au and au.
Figure 3: The same as Fig. 1 for Brangane asteroid family data from Nesvorný et al. (2015). (Top panel) is equal to au and , is equal to au. (Bottom Panel) is plotted with , = 2.584 au and au.

2.2 Contribution to width of young asteroid family V-shapes by the Yarkovsky effect

The value obtained by the V-shape , and determination method in Section 2.1 includes the contribution of the initial ejection Velocity field from Eq. 7 and the contribution to from the Yarkovsky effect (Vokrouhlický et al., 2006a; Nesvorný et al., 2015)

(10)

where is the width of the V-shape due to the Yarkovsky effect and is the width of the V-shape due to the inital ejection velocity of fragments. Asteroid family V-shapes with only in Eq. 10 are indistinguishable from asteroid families which have contribution to their value of from both the Yarkovsky effect and ejection velocity. The nominal value of exceed more than 50 of for asteroid families younger than 100 Myrs (Nesvorný et al., 2015; Carruba and Nesvorný, 2016). The error on the parent body size and the resultant calculation of from the parent body’s escape velocity can be large enough so that there is a possibility that . In this work, we select for analysis all the families for which , considering that these families are young enough that the contribution of the Yarkovsky effect to the spread in of the family fragments is minimal and therefore we assume that the value of obtained with the techniques in Section 2.1 is equal to .

However, it should be noted that the Yarkovsky effect still affects the displacement in of even young family members. Nesvorný and Bottke (2004) and Carruba et al. (2016b) showed that the Yarkovsky drift rate and displacement in could be determined for Karin family fragments by backwards integrating the orbits of the family fragments backwards in time and measuring the convergence of the ascending node, , and longitude of perihelion, , of the Karin family fragments relative to asteroid Karin while under the influence of the Yarkovsky effect. Although the effect of the Yarkovsky force on the Karin family fragments is strong enough to be detected using the techniques in Nesvorný and Bottke (2004) and Carruba et al. (2016b), the displacement in over the age of the young Karin family is not large enough to affect our assumptions (see Section A.7 about the Karin family as an example).

2.3 Data set and uncertainties of measurements

2.3.1 Data set

The data used to measure the V-shapes of asteroid family were taken from the MPC catalogue for the magnitudes. Family definitions were taken from Nesvorný et al. (2015). Asteroid family data for the 1993 FY, Brangane, and Iannini families were used from both Milani et al. (2014) and Nesvorný et al. (2015) to verify that the V-shape technique provided similar results for , and for the same family with asteroid data taken from different asteroid family databases. Results from the V-shape technique using asteroid family data from Nesvorný et al. (2015) were repeated with asteroid family data from Milani et al. (2014) for the 1993 FY, Aeolia, Brangane, Brasilia, Iannini and König asteroid families as seen for the Brangane family in Figs. 3 and 4. Family visual albedo, , data from Masiero et al. (2013) and Spoto et al. (2015) were used to calibrate the conversion from magnitudes to asteroid using the relation (Bowell et al., 1988) where (Pravec and Harris, 2007). Numerically and analytically calculated MBA proper elements were taken from the Asteroid Dynamic Site222http://hamilton.dm.unipi.it/astdys/ (Knežević and Milani, 2003). Numerically calculated proper elements were used preferentially and analytical proper elements were used for asteroids, that had numerically calculated elements as of April 2017.

Figure 4: Same as Fig. 3, but repeated for the Brangane family defined by Milani et al. (2014)

2.3.2 Uncertainty of

The value of located where peaks in vs. space represents the best estimate of the of a asteroid family’s V-shape using the nominal and asteroid values. Different values in the physical properties of asteroids cause a spread in possible values that measured together are the uncertainty in the measured value of . Changes in asteroids’ is caused by variations in their magnitude measurements and spread in during the conversion of asteroid to . In addition to the variety of different possible values for asteroids and a lack of complete information about the true population of asteroids within a family, the contribution of outliers to a family’s vs. distribution can increase the spread in values compatible with the family V-shape. We devise the following Monte Carlo procedure to quantify the spread in measurements of family V-shapes caused by random differences in the asteroid physical properties between family members and incomplete information about the asteroid family member population.

At least 1,200 Monte Carlo trials are completed per family. Some families have significantly more than 1,200 Monte Carlo trials as described in the Appendix if additional CPU time was available. In each trial, the location of the peak value in vs. is recorded. Three steps are completed to randomise the asteroid family data from the original vs. distribution per trial. The first step is to create a resampled data set of family fragments by removing objects randomly where is the number of objects in vs. space to include variations caused by incomplete knowledge of the asteroid family fragment population. Incompleteness of asteroid family fragments increases for smaller fragments and is more pronounced in the middle and outer portions of the main belt (Jedicke and Metcalfe, ; Jedicke et al., 2002). The variation of caused by the incomplete knowledge of the family fragment population is more weighted towards smaller fragments than larger fragments as a result of the increased incompleteness and greater number of smaller Main Belt asteroids in the asteroid family catalogues

A second step is taken to determine the variation caused by incomplete information in the family fragment population by resampling the fragments’ by their own distribution per bin. In this step, family fragments are randomised by the semi-major axis distribution of fragments in each bin with a size of 0.001 km.

The third step is to randomise the measurements of and of the asteroids by their known uncertainties. Asteroid values were randomised between 0.2 and 0.3 magnitudes known uncertainties for values from the Minor Planet Center Catalogue (Oszkiewicz et al., 2011; Pravec et al., 2012) with an average offset of 0.1 magnitudes consistent for asteroids with (Pravec et al., 2012; Vereš et al., 2015). After the values are randomised, asteroid fragments’ were converted to using the relation

(11)

from Harris and Lagerros (2002), and a value of chosen at random for each asteroid using central values and uncertainties per asteroid family from Masiero et al. (2013) and Spoto et al. (2015).

The mean and root mean square (RMS) uncertainty of was determined from the distribution of in the Monte Carlo trials. Having more fragments and a well defined-V-shape causes the Monte Carlo technique to produce a narrower distribution in (E.g.,  for the Karin family, , Fig. 5), while having fewer fragments and a more diffuse V-shape results in a broader distribution (e.g., for the 1993 FY family, , Fig. 6).

Figure 5: Histogram of located at the peak value of to in each of the 1,200 trials repeating the V-shape technique for the Karin family. The mean of the distribution is centered at = 0.97 0.05 and the bin size in the histogram is 0.04 consistent with Nesvorný et al. (2002).
Figure 6: The same as Fig. 5 with 4,900 trials repeating the V-shape technique for the 1993 FYfamily. The mean of the distribution is centered at = 1.03 0.11 and the bin size in the histogram is 0.06.

3 Results

3.1 Synthetic family

It is generally expected that 1 as discussed in Section 2. Recent work on the V-shapes of asteroid families 100 Myrs-old suggests that 0.8 (Bolin et al., 2017, ). The time it takes to transition from a V-shape having its equal to to equal to is the time it takes for families to have their initial ejection velocity fields erased by the Yarkovsky effect. We determine the time when the Yarkovsky effect erases by simulating the initial ejection of fragments from the disruption of a parent body and their subsequent spreading caused by the Yarkovsky effect.

The break up of a synthetic asteroid family and its fragments’ subsequent evolution due to the Yarkovsky effect is simulated by using 650 particles at and distributed in vs. space according to Eq. 2 with and = 30 using fragments with distributed according to the known members of the Erigone family defined by Nesvorný et al. (2015). The eccentricity and inclination distributions were determined by using Gaussian scaling described in Zappalà et al. (2002). = 30 corresponds to a typical initial displacement of au for a 5 km diameter asteroid. The Yarkovsky drift rates were defined with

(12)

from Spoto et al. (2015); Bolin et al. (2017) with 4.7 , 2.37 au, 0.2, = 5 km, = 2.5 and bond albedo, , is equal to 0.1, surface conductivity between 0.001 and 0.01 and 0 (Bottke et al., 2006; Vokrouhlický et al., 2015). For the synthetic family, = 2.3 , and is uniformly distributed between -1 and 1. An = 0.8 was chosen as measurements of asteroid families old enough to have their fragments significantly modified by the Yarkovsky effect have 0.7 0.9 (Bolin et al., 2017). The particles were evolved with the Yarkovsky effect and gravitational perturbations from Mercury, Venus, Earth, Mars, Juputer and Saturn using the _ code (Levison and Duncan, 1994). Particles are removed from the simulation if they collide with one of the planets or evolve on to small perihelion orbits. YORP rotational and spin-axis variation are not included in the simulation.

The V-shape identification technique was applied on the synthetic family data at Time = 0 by using the techniques in Section 2.1. Eqs. 8 and 9 are integrated using the interval (,) for the Dirac delta function and the interval [] for the Dirac delta function . Eq. 6 is truncated to 0.04 for and to 0.60 for 0.60 . Asteroids with 0.04 0.60 were chosen because the number of asteroids in this is large enough so that the leading edge of the V-shape is defined by asteroids with 1.0 or -1.0 according to Eq. 2.

The V-shape identification technique located a peak at as seen in the top panel of Fig. 1. The peak value of is 11 standard deviations above the mean value of in the range 2.35 au 2.38 au, 1.0 au 6.0 au and 0.8 1.2. A au was used. The concentration of the peak to one localized area in vs. space is due to the sharpness of the synthetic family’s V-shape border.

The V-shape identification technique was applied to family fragments at 1 Myr steps for the first 100 Myrs of the simulation. The value of for the V-shape linearly decreases below 1.0 and reaches 0.8, equal to after 20 Myrs (see Fig. 7). There is a steep drop in from 1.0 to 0.97 in the first 1-2 Myrs of the simulation that is possibly due to the fragments near the borders of the family becoming spread in according to Eq. 12. This is because fragments at the border of the family with cos() = 1 are the Yarkovsky front-runners and cause a very quick spreading of the family, rapidly changing the value of . Assuming equal to 0.8 for the Yarkovsky drift size dependence in the Main Belt, asteroid family V-shapes with a measured closer to 0.8 reveal that the dispersion of the family is dominated by the Yarkovsky effect over the initial ejection of fragments and older than Myrs. Family V-shapes that have measured values of 1.0 have the contribution of the initial ejection velocity field dominant to the value of and the measured value of is characteristic of the size-dependece of the initial ejection velocity.

Figure 7: Time vs. for the first 30 Myrs of time evolution of the fragments of a synthetic asteroid family given an 1.0 according to Eq. 2 and evolved in time 0.8 according to Eq. 12. The steep decrease in after 0 Myrs is due to the randomization of asteroid fragment obliquity and subsequent evolution of their ’s according to Eq. 12. The dark line is a linear fit to the first 20 Myrs of simulation data.

3.2 Young asteroid family ejection velocity V-shapes

The V-shape identification technique was applied to 11 asteroid families noted for their young ages between a few Myrs to a few 10 Myrs (Spoto et al., 2015; Nesvorný et al., 2015) listed in the first column of Table 2, selected for having as explained in Section 2.2. The measured value of their V-shape’s and uncertainties determined by the techniques in Sections 2.1 and 2.3.2 as well as physical properties used to measure the of family V-shapes are summarized in Table 2. A description of how the V-shape identification technique is implemented for each family is described in the Appendix.

Variable Description
Asteroid diameter in km
Semi-major axis in au.
Eccentricity.
Inclination in degrees.
Reciprocal of the diameter, in km.
The location of the V-shape centere in au.
Mean motion in
Ejection velocity in .
The of an initial velocity V-shape.
Visual albedo.
Total V-shape width in au.
defined by Eq. 6.
Number density of objects between the nominal and outer V-shapes.
Number density of objects between the nominal and inner V-shapes.
Difference in between the nominal and outer/inner V-shapes.
Absolute magnitude.
V-shape width due to Yarkovsky spreading of fragments in au.
V-shape width due to the initial ejection of fragments in au.
Number of family members used with the V-shape technique.
The of a Yarkovsky V-shape.
Asteroid density in .
Bond albedo.
Asteroid obliquity.
Table 1: Description of variables in order of appearance.
Designation Tax. D -
(km) ( au) ( au) (au) (km)
1993 FY S 15 7.5 8.1 87 2.847 1.03 0.11 0.17 0.05 2.0 - 4.7
Aeolia X 35 7.7 3.6 225 2.7415 1.0 0.07 0.11 0.03 1.3 - 3.3
Brangane S 42 7.2 3.9 171 2.584 0.95 0.04 0.1 0.03 1.5 - 7.5
Brasilia X 34 13.4 15.2 548 2.855 1.0 0.1 0.24 0.07 1.0 - 7.0
Clarissa C 39 3.9 3.8 179 2.404 0.95 0.04 0.06 0.02 1.1 - 8.9
Iannini S 14 5.6 2.2 129 2.644 0.97 0.07 0.30 0.10 0.9 - 3.7
Karin S 40 12.1 3.1 429 2.865 0.97 0.05 0.21 0.06 1.8 - 10.3
König C 37 3.3 4.4 315 2.574 0.91 0.03 0.06 0.01 1.4 - 5.5
Koronis(2) S 58 11.1 2.3 235 2.869 1.09 0.05 0.14 0.04 1.1 - 3.6
Theobalda C 97 10.7 9.0 349 3.179 0.95 0.04 0.06 0.02 2.1 - 15.2
Veritas C 124 21.0 12.2 1135 3.168 1.01 0.04 0.07 0.02 3.2 - 31.7
Table 2: The measured value of of young asteroid families: Asteroid family fragment taxonomies are taken from Nesvorný et al. (2003); Willman et al. (2008); Harris et al. (2009); Molnar and Haegert (2009); Novaković et al. (2010); Spoto et al. (2015); Nesvorný et al. (2015). Diameters for the parent body, , were taken from the means of asteroid family parent bodies in Brož et al. (2013) and Durda et al. (2007) if was available from both sources. for Iannini was taken from Nesvorný et al. (2003). The of the Koronis(2)and Veritas families were determined with techniques from Nesvorný et al. (2015) to estimate parent body size. values of asteroid family members are taken from Masiero et al. (2013) and Spoto et al. (2015). is the number of family fragments used in the determination of a family’s V-shape. and are the boundaries of the of the smallest and largest fragments used to measure an asteroid family v-shape’s .

4 Discussion and Conclusion

We have demonstrated that the techniques of Bolin et al. (2017) can be used not only to identify asteroid family V-shapes, but also to measure the spreading of family fragments caused by the initial ejection velocity field from the disruption of the parent body and by the Yarkovsky effect. We have demonstrated, following the work of Vokrouhlický et al. (2006a), that the functional form of the spread of a family created entirely initial ejection field of fragments from their parent body’s disruption (i.e.with = 0 in Eq. 10), and the spread caused by the Yarkovsky effect (i.e.with 0 and 0 in Eq. 10) are functionally equivalent.

We have measured the V-shapes of 11 young (100 My-old) asteroid families located within the inner, central and outer MB and we have found that all of them 1.0. We associate this value of to the initial velocity V-shape’s , concluding that the initial ejection velocity is proportional to , as was already assumed, because these families are too young to have been substantially modified by the Yarkovsky effect. Our measurements were repeated for each of the 11 families using the Monte Carlo scheme described in Section 2.3.2 and found that the 1 uncertainty of the trial measurements were within of the mean trial value for most of the families. Some of the families such as the Aeolia, Clarissa, Brangane, Iannini and Koronis(2) families had slightly skewed trial distributions in the positive and negative directions.

The average value of of the 11 family V-shapes in this study is 0.97 0.02, or within in the Student’s t-distribution 99.8 confidence interval 0.95 - 0.99. Additionally, our average measurement of 1.0 confirms the results of laboratory and numerical experiments of asteroid disruption events showing 1.0 Fujiwara et al. (1989); Michel et al. (2001); Nesvorný and Bottke (2004). Other studies focusing on modelling the observed distribution of family fragments with disruption simulations such as for the Karin family (Nesvorný et al., 2002, 2006) or the distribution of family such as for the Koronis family (Carruba et al., 2016a) also show that 1.0.

The determination technique in Section 2.1 can be used to determine whether or not an asteroid family V-shape is young enough to not have been significantly altered by the Yarkovsky effect. There is indication that the value of for Yarkovsky V-shapes, or , for older families whose fragments have been significantly modified by the Yarkovsky effect as described by Eq. 6 is between 0.7 - 0.9 due to possible thermal inertia dependence on asteroid size and its effect on the Yarkovsky drift rate as a function of asteroid size (Delbo et al., 2007, 2015; Bolin et al., 2017). Asteroid family V-shapes that have been significantly affected by the Yarkovsky effect will have values as described in Eq. 6 that are closer to inside the range of 0.7 - 0.9.

In fact, the ages of some of these families have been determined by the use of alternative methods such as backwards integrating the orbits of selected bodies in the families or by modedlling the diffusion of fragments caused by chaos Nesvorný et al. (2002, 2003); Tsiganis et al. (2007); Novaković et al. (2010); Carruba et al. (2017). By these methods, Iannini, Karin, Theobalda and Veritas all have ages between 5 and 9 Myrs independently ruling out significant modification of the Yarkovsky effect on their fragments’ .

We demonstrate that V-shapes in vs space created purely by the initial ejection of fragments can be separated from those that are created by a combination of the initial ejection velocity of fragments and the Yarkovsky effect using the measurement of as generically described for asteroid family V-shapes by Eq. 6. We assume that the value of 0.8 as indicated as indicated by Bolin et al. (2017) for very old families which have lost memory of their initial dispersion. We find that the time-scale for to reach as a result of the modification of fragments’ by the Yarkovsky effect to is on the order of 20 Myrs as seen in Fig. 7. Interestingly, the backwards integration technique is unable to identify families and determine accurate family ages for families older than 20 Myrs (Nesvorný et al., 2003; Radović, 2017).

The V-shape identification method measuring family V-shapes’ provides a way to distinguish whether a family V-shape is caused by a combination of the initial ejection of fragments and the Yarkovsky effect, or only due to initial ejection of fragments. The measurement of asteroid family V-shape provides an additional independent evidence of the subsequent orbital evolution of asteroid family fragment due to the Yarkovsky effect after the initial placement of fragments due to the initial ejection of fragments because asteroid families old enough to have their fragments’ orbits modified by the Yarkovsky will have an 1.0 compared to the case where asteroid family fragments’ orbits remain unmodified after their parent’s disruption where their family V-shape would have = 1.0.

It may be possible to independently constrain the degree to which an asteroid family fragments’ spread in has been modified in part by the Yarkovsky effect relative to the spread caused by the initial ejection of fragments. Family V-shapes with a higher proportion of relative to in their total value of may have a higher 1 compared to family V-shapes with a higher proportion of relative to in their total value of . Distinguishing the families with higher proportion of compared to requires measurements of with small uncertainties. Using methods of removing outliers by colours and other physical data such as the method of Radović (2017) may improve the precision of measurements independently of other methods such as the V-shape criterion of Nesvorný et al. (2015) which may bias the measurement of a V-shape’s towards the assumption of used in the V-shape criterion before the actual measurement of is made. Understanding more about the evolution of family fragments’ and its affect on the measurement of may provide an independent constraint on a family’s age provided the physical properties of the asteroid family fragments are known such as density and surface thermal conductivity.

Acknowledgments

We would like to thank the reviewer of our manuscript, Valerio Carruba, for providing helpful comments and suggestions for improving the quality of the text. BTB is supported by l’Ècole Doctorale Sciences Fondatementales et Appliquées, ED.SFA (ED 364) at l’Université de Nice-Sophia Antipolis. KJW was supported by the National Science Foundation, Grant 1518127. BTB would like to acknowledge James W. Westover for thought-provoking discussions on the implementation of large-scale computing resources and algorithms that were used in the completion of this work.

Appendix A Appendix

a.1 1993 Fy

The 1993 FYasteroid family located in the outer Main Belt was first identified by Nesvorný (2010) and consists of mostly S-type asteroids (Spoto et al., 2015). The age of the family is roughly estimated to be 200 Myrs by Brož et al. (2013) and Myrs by Spoto et al. (2015). It should be noted that ages of asteroid families by Spoto et al. (2015) are upper limits to the family age because they are computed with the assumption that 0. The V-shape identification technique was applied to 87 asteroids belonging to the 1993 FYasteroid family defined by Nesvorný et al. (2015). Eqs. 8 and 9 are integrated using the interval (,] for the Dirac delta function because the inner half of the family V-shape for 1993 FYis more densely populated than the outer V-shape half possibly due to lack of completeness of 1993 FYfamily members as seen in the bottom panel of Fig. 8. The interval [] for the Dirac delta function to avoid the potential distortion of the family V-shape because of smaller fragments interacting with the 5:2 MMR at 2.81 au. Eq. 6 is truncated to 0.21 for and to 0.49 for 0.49. Asteroid values were converted to using Eq. 11 using the value of = 0.184 typical for members of the 1993 FYfamily (Spoto et al., 2015).

Figure 8: The same as Fig. 1 for 1993 FYasteroid family data from Nesvorný et al. (2015). (Top panel) is equal to au and , is equal to au. (Bottom Panel) is plotted with . = 2.847 au and au.

The peak in at as seen in the top panel of Fig. 8 is located in the range 2.83 au 2.86 au, 6.0 au 1.1 au and 0.8 1.2. A au was used. The technique was repeated with the 1993 FYfamily defined by Spoto et al. (2015) resulting in identical results.

The V-shape identification technique was repeated in 4,900 Monte Carlo runs where magnitudes were randomised by the typical magnitude uncertainty of 0.25 for asteroids in the MPC catalogue (Oszkiewicz et al., 2011; Vereš et al., 2015) and their was assumed to be the average value of for family fragments in the 1993 FYfamily fragments of 0.18 with an uncertainty of 0.04 (Spoto et al., 2015). The value of in 4,900 Monte Carlo trials ranges between 0.5 1.5 and is on average 1.0 with a RMS uncertainty of 0.11 as seen in Fig. 6. The large RMS uncertainty of 0.11 is due to the low number of asteroids used to measure the family V-shape’s . The value of 1.0 and a similar value of compared to calculated using Eq. 7 assuming = 15 from Brož et al. (2013) suggests that the spread of fragments in the 1993 FYfamily in vs. space is almost entirely due to the ejection velocity of the fragments with minimal modification in due to the Yarkovsky effect.

a.2 Aeolia

The X-type Aeolia asteroid family located in the outer Main Belt was first identified by Nesvorný (2010). The age of the family is estimated to be only 100 Myrs by Brož et al. (2013) and Myrs by Spoto et al. (2015). The V-shape identification technique was applied to 225 asteroids belonging to the Aeolia asteroid family defined by Nesvorný et al. (2015). Eqs. 8 and 9 are integrated using the interval (,] for the Dirac delta function because the outer half of the family V-shape for Aeolia has a less defined border than the inner V-shape half due to possible interlopers as seen in the bottom panel of Fig. 9. The V-shape criterion of Nesvorný et al. (2015) was not used to remove potential interlopers because it assumes a functional form of in Eq. 6 which would result in artificially trimming and biasing the family V-shape towards having an . The interval [] for the Dirac delta function was used to remove at higher values on the inner edge of the family V-shape in the application of the V-shape identification technique. Eq. 6 is truncated to 0.31 for and to 0.78 for 0.78. Asteroid values were converted to using Eq. 11 using the value of = 0.11 typical for members of the Aeolia family (Spoto et al., 2015).

Figure 9: The same as Fig. 1 for Aeolia asteroid family data from Nesvorný et al. (2015). (Top panel) is equal to au and , is equal to au. (Bottom Panel) is plotted with , = 2.742 au and au.

The peak in at as seen in the top panel of Fig. 9 is located in the range 2.73 au 2.75 au, 2.0 au 4.0 au and 0.8 1.2. A au was used. The technique was repeated with the Aeolia family defined by Spoto et al. (2015) resulting in similar results. In addition, the Aeolia family is noted by Spoto et al. (2015) for having an asymetrical V-shape where the outer half the V-shape has a steeper slope. The , , technique was repeated using [,) for the Dirac delta function resulting in a peak located at at . A lower value of of for the outer V-shape half compared to the value of of is in is in agreement with Spoto et al. (2015) for the outer V-shape half of the Aeolia family having a smaller value of .

1,405 Monte Carlo runs were completed where magnitudes were randomised by the typical magnitude uncertainty of 0.25 for asteroids in the MPC catalogue (Oszkiewicz et al., 2011; Vereš et al., 2015) and their was assumed to be 0.11on average with an uncertainty of 0.03 (Spoto et al., 2015). The value of in 1,405 Monte Carlo trials is on average 1.0 with a RMS uncertainty of 0.07 as seen in Fig. 10. The large RMS uncertainty of 0.07 is due to the low number of asteroids used to measure the family V-shape’s on the inner half of the family V-shape. The distribution of Monte Carlo runs is slightly positively skewed such that the most probable value is slightly lower than the mean of . The value of 1.0 and a smaller value of au compared to au calculated from Eq. 7 assuming = 22 , the escape velocity from a parent body with a parent body diameter, km (Durda et al., 2007; Brož et al., 2013) and typical for X-type asteroids (Carry, 2012) suggests that the spread of fragments in the Aeolia family in vs. space is due to the ejection velocity of the fragments with minimal modification in due to the Yarkovsky effect.

Figure 10: The same as Fig. 5 with 1,930 trials repeating the V-shape technique for the Aeolia family. The mean of the distribution is centered at = 1.0 0.07 and the bin size in the histogram is 0.05.

a.3 Brangane

The S-type Brangane asteroid family located in the central Main Belt was first identified by Nesvorný (2010). The age of the family is estimated to be 5040 Myrs by Brož et al. (2013) with a similar estimate in Spoto et al. (2015). The V-shape identification technique was applied to 171 asteroids belonging to the Brangane asteroid family defined by Nesvorný et al. (2015). Eqs. 8 and 9 are integrated using the interval (,] for the Dirac delta function because the outer half of the family V-shape for Brangane has a fewer asteroids in vs, space on its outer V-shape half as seen panel of Fig. 3. The interval [] for the Dirac delta function was used to cover the range of the entire inner half of the Brangane family’s V-shape. Eq. 6 is truncated to 0.19 for and to 0.92 for 0.92. Asteroid values were converted to using Eq. 11 using the value of = 0.10 typical for members of the Brangane family (Masiero et al., 2013; Spoto et al., 2015).

The peak in at as seen in the top panel of Fig. 3 is located in the range 2.57 au 2.60 au, 3.5 au 5.5 au and 0.7 1.3. A au was used. The technique was repeated with the Brangane family defined by Spoto et al. (2015) resulting in similar results as seen in Fig. 4. The , , technique was repeated using [,) for the Dirac delta function resulting in a peak located at at . A lower value of of for the outer V-shape half compared to the value of of is in is in agreement with Spoto et al. (2015) for the outer V-shape half of the Brangane family having a smaller value of .

1,400 Monte Carlo runs were completed where magnitudes were randomised by the typical magnitude uncertainty of 0.25 for asteroids in the MPC catalogue (Oszkiewicz et al., 2011; Vereš et al., 2015) and their was assumed to be 0.10 with an uncertainty of 0.03 (Masiero et al., 2013). The value of in 1,400 Monte Carlo trials is on average 0.95 with a RMS uncertainty of 0.04 as seen in Fig. 11. The distribution of Monte Carlo runs is slightly negatively skewed such that the most probable value is slightly higher than the mean of . The value of 1.0 and a smaller value of au compared to au calculated from Eq. 7 assuming = 23 , the escape velocity from a parent body with a parent body diameter, km (Brož et al., 2013) and typical for S-type asteroids (Carry, 2012) suggests that the spread of fragments in the Brangane family in vs. space is mostly due to the ejection velocity of the fragments with only moderate modification in due to the Yarkovsky effect.

Figure 11: The same as Fig. 5 with 1,400 trials repeating the V-shape technique for the Brangane family. The mean of the distribution is centered at = 0.95 0.04 and the bin size in the histogram is 0.03.

a.4 Brasilia

The X-type fragment Brasilia asteroid family first identified by Zappalà et al. () located in the outer Main Belt. The M/N solar system dust band was later attributed to the formation of the family with the asteroid 293 Brasilia being an interloper in its own family (Nesvorný et al., 2003; Brož et al., 2013). The asteroid 1521 Sejnajoki is more likely to be the largest asteroid family member in the Brasilia asteroid family, but we will use Brasilia as the name for the asteroid family. The age of the Brasilia family is estimated to be 5040 Myrs by Brož et al. (2013). The V-shape identification technique was applied to 584 asteroids belonging to the Brasilia asteroid family defined by Nesvorný et al. (2015). Eqs. 8 and 9 are integrated using the interval [,) for the Dirac delta function because the inner half of the family V-shape for Brasilia is clipped due to the presence of the 5:2 MMR at 2.81 au as seen in the bottom panel of Fig. 12. The interval [] for the Dirac delta function was used to cover the majority of the range of the outer V-shape half while excluding possible interlopers at larger values of . Eq. 6 is truncated to 0.13 for and to 0.53 for 0.53. Asteroid values were converted to using Eq. 11 using the value of = 0.24 typical for members of the Brasilia family (Masiero et al., 2013; Spoto et al., 2015).

Figure 12: The same as Fig. 1 for Brasilia asteroid family data from Nesvorný et al. (2015). (Top panel) is equal to au and , is equal to au. (Bottom Panel) is plotted with , = 2.855 au and au.

The peak in at as seen in the top panel of Fig. 12 is located in the range 2.83 au 2.89 au, 5 au 2.0 au and 0.8 1.2. A au was used. The technique was repeated with the Brasilia family defined by Spoto et al. (2015) resulting in similar results.

1,500 Monte Carlo runs were completed where magnitudes were randomised by the typical magnitude uncertainty of 0.25 for asteroids in the MPC catalogue (Oszkiewicz et al., 2011; Vereš et al., 2015) and their was assumed to be 0.24 with an uncertainty of 0.06 (Masiero et al., 2013). The value of in 1,500 Monte Carlo trials is on average 0.99 with a RMS uncertainty of 0.1 as seen in Fig. 13. The value of 1.0 and a similar value of au compared to au calculated from Eq. 7 assuming = 22 , the escape velocity from a parent body with a parent body diameter, km (Brož et al., 2013) and typical for X-type asteroids (Carry, 2012) suggests that the spread of fragments in the Brasilia family in vs. space is mostly due to the ejection velocity of the fragments with only moderate modification in due to the Yarkovsky effect.

Figure 13: The same as Fig. 5 with 1,500 trials repeating the V-shape technique for the Brasilia family. The mean of the distribution is centered at 1.0 0.10 and the bin size in the histogram is 0.07.

a.5 Clarissa

The C complex Clarissa asteroid family located in the inner Main Belt was first identified by Nesvorný (2010). The age of the family is estimated to be Myrs by Brož et al. (2013). The V-shape identification technique was applied to 179 asteroids belonging to the Clarissa asteroid family defined by Nesvorný et al. (2015). Eqs. 8 and 9 are integrated using the interval (,] for the Dirac delta function because the outer half of the family V-shape for Clarissa has a fewer asteroids in vs, space on its outer V-shape half as seen panel of Fig. 14. The interval [] for the Dirac delta function was used to cover the full range of fragments in the Clarissa family inner half V-shape. Eq. 6 is truncated to 0.11 for and to 0.91 for 0.91. Asteroid values were converted to using Eq. 11 using the value of = 0.06 typical for members of the Clarissa family (Masiero et al., 2013).

Figure 14: The same as Fig. 1 for Clarissa asteroid family data from Nesvorný et al. (2015). (Top panel) is equal to au and , is equal to au. (Bottom Panel) is plotted with , = 2.404 au and au.

The peak in at as seen in the top panel of Fig. 14 is located in the range 2.38 au 2.43 au, 3.0 au 5.0 au and 0.8 1.1. A au was used.

4,700 Monte Carlo runs were completed where magnitudes were randomised by the typical magnitude uncertainty of 0.25 for asteroids in the MPC catalogue (Oszkiewicz et al., 2011; Vereš et al., 2015) and their was assumed to be 0.06 with an uncertainty of 0.02 (Masiero et al., 2013). The value of in 4,700 Monte Carlo trials is on average 0.95 with a RMS uncertainty of 0.04 as seen in Fig. 15. The distribution of Monte Carlo runs is slightly negatively skewed such that the most probable value is slightly higher than the mean of . The value of 1.0 and a similar value of au compared to au calculated from Eq. 7 assuming = 17 , the escape velocity from a parent body with a parent body diameter, km (Brož et al., 2013) and typical for C-type asteroids (Carry, 2012) suggests that the spread of fragments in the Clarissa family in vs. space is mostly due to the ejection velocity of the fragments with only moderate modification in due to the Yarkovsky effect.

Figure 15: The same as Fig. 5 with 4,700 trials repeating the V-shape technique for the Clarissa family. The mean of the distribution is centered at = 0.95 0.04 and the bin size in the histogram is 0.07.

a.6 Iannini

The S-type Iannini asteroid family located in the central Main Belt and the presence of the J/K solar system dust band is attributed to the formation of the family (Nesvorný et al., 2003; Willman et al., 2008). The age of the Iannini family is estimated to be 55 Myrs by Brož et al. (2013). The V-shape identification technique was applied to 584 asteroids belonging to the Iannini asteroid family defined by Nesvorný et al. (2015). Eqs. 8 and 9 are integrated using the interval (,) for the Dirac delta function and the interval [] is used for the Dirac delta function to contain fragments defining the border of the Iannini family V-shape. Eq. 6 is truncated to 0.4 for and to 1.5 for 1.5. Asteroid values were converted to using Eq. 11 using the value of = 0.32 typical for members of the Iannini family (Masiero et al., 2013).

The technique was repeated with the Iannini family defined by Spoto et al. (2015). The Iannini family defined by Spoto et al. (2015) differs from the definition of the Iannini family by Nesvorný et al. (2015) by using the asteroid Nele as the largest fragment in the family. If the 17 km diameter asteroid Nele is considered to be the largest remaining fragment of the Iannini family, the family V-shape is slightly asymmetric Spoto et al. (2015). Regardless of the potential asymmetric V-shape of the Iannini/Nele family, the V-shape has a symmetrical shape in both Spoto et al. (2015) and Nesvorný et al. (2015) definitions of the Iannini family in the interval . This interval does not include asteroids Iannini and Nele which have equal to 0.2 km and 0.06 km respectively, and the results of the , and determination technique are similar when applied to both catalogues. An alternative explanation to the membership of Nele to the Iannini family is that it is an interloper because of how far it is offset from the apex of the main family V-shape (Nesvorný et al., 2015).

Figure 16: The same as Fig. 1 for Iannini asteroid family data from Nesvorný et al. (2015). (Top panel) is equal to au and , is equal to au. (Bottom Panel) is plotted with , = 2.644 au and au.

The peak in at as seen in the top panel of Fig. 16 is located in the range 2.637 au 2.65 au, 0.5 au 3.5 au and 0.7 1.3. A au was used. The technique was repeated with the Iannini family defined by Spoto et al. (2015) resulting in similar results.

1,400 Monte Carlo runs were completed where magnitudes were randomised by the typical magnitude uncertainty of 0.25 for asteroids in the MPC catalogue (Oszkiewicz et al., 2011; Vereš et al., 2015) and their was assumed to be 0.32 with an uncertainty of 0.1 (Masiero et al., 2013). The value of in 1,600 Monte Carlo trials ranges between 0.5 1.5 and is on average 0.97 with a RMS uncertainty of 0.07 as seen in Fig. 17. The distribution of Monte Carlo runs is slightly positively skewed such that the most probable value is slightly lower than the mean of . The value of 1.0 and a smaller value of au compared to au calculated from Eq. 7 assuming = 5.6 , the escape velocity from a parent body with a parent body diameter, km (Nesvorný et al., 2015) and