The minimal sizes of impactors that formed the Vesta family and 15 other asteroid families

Abstract

The minimal sizes of impactors that produced asteroid families were calculated, and their maximal sizes, i.e. the size that corresponds to head-on collisions, were estimated. We used data for 16 large families and the physical parameters of some members of these families available in NASA's JPL updated in 2022. We found that the typical minimal sizes of family-forming impactors were of the order of one-tenth the size of the parent bodies (PBs). The Themis family, for which the ratio of the radii r_imp/R_PB = 0.08 and the mass loss is as large as 0.55, presents an example. The families of Juno and Euphrosyne, with the impactor to PB size ratio of an order of 0.01, are rather exceptional. The PBs of Juno and Euphrosyne families lost only 0.0014 and 0.0061 of their mass, respectively. It was found that the double structure of the Vesta family could have originated from two impacts by the bodies with radii as small as about 4 km, at least.

minor planets, asteroids: general

1 INTRODUCTION

While some asteroid families (AFs) may be formed by the break-ups of parent bodies (Carruba et al. 2018, 2020; Novaković et al. 2022), most AFs originated from mutual collisions of asteroids. Many papers have recently focused on AFs population properties, dynamical evolution with or without the Yarkovsky effect, and extrapolating results of collision experiments from laboratory scale to asteroid scale. However, not much attention has been given to estimating the possible size of impactors that formed known asteroid families. The present state of many AFs identified to date is known. That means that there are known AF populations, that is the numbers of the asteroid families’ members (AFMs). For many AFs, the number of members is approaching the current limit of observational detection. In this paper as well as in the previous one (Leliwa-Kopystynski & Wlodarczyk 2020) the highest value of the absolute magnitude of AFMs was ∼18–20, see the Appendix. While the orbital parameters of most AFMs are known with high accuracy, physical properties like sizes, albedos, and the mean densities of most of them are usually known very poorly. That concerns especially the sizes, the albedos, and the mean densities, which are essential in this work. The physical properties of the largest member (LM) of an AF are usually known much better than the physical properties of other members of that family. The latter are usually considerably smaller than the LM. The aim of this paper are estimations of minimal sizes of impactors that formed some AFs.

There are many papers on AFs. They particularly deal with the AFs populations, with AFs evolutions forward and backward (without or with Yarkovsky effect), and with AFs ages. There are as well a lot of experimental papers aimed at extrapolation their results from the laboratory scale to the asteroid scale. The most recent review paper compacting knowledge concerning AFs is probably that of Novaković et al. (2022). However, in their exhaustive review paper concerning AFs they are not dealing with the impactors that formed them. The terms ‘impactor’ and ‘projectile’ appear only few times but ‘impactor size’ is not present at all. The same concerns other prominent papers, like Milani et al. (2018), which do not mention impactors. The presentations that one of us (JLK) had the pleasure to follow recently at the Asteroid Comets Meteors conference (Flagstaff, US, June 2023) did not consider the size of impactors in connection with the formation of AFs.

Previous works estimated this parameter based on the size–frequency distributions (SFDs) applied to hits of an assumed impactor onto an assumed target. Here ‘assumed’ means that the sizes and densities of both bodies involved are known. The impact angle as well as the speed of collision are the parameters. Asphaug (1997) obtained that a ∼42-km-diameter asteroid striking Vesta's basaltic crust formed the Vesta family. Note, that the Asphaug's estimate originated from 15 yr before the spacecraft Dawn entered orbit around Vesta (2011 July 16). Nesvorný et al. (2006) studied the Karin cluster. After 30 simulations using SPH 3D code and after comparing each of them with the observed SFD of the Karin family they concluded that that family was formed by an impact of ≃5.8 km projectile onto ≃33 km parent body with ≃6–7 km s⁻¹ speed and ≃45° impact angle. Results presented above allow for a rough estimate of the mass ratio of the impactor to the PB as being equal to (42/262)³ = 0.0041 for Vesta, and (5.8/33)³ = 0.0054 for Karin. (In this paper in numerical estimates it is assumed that the target and impactor are spherical and their densities are the same, if not otherwise stated.) Anticipating the results of this work presented further on we should note that the lower limits of the radii of the impactors that were able to form the Vesta family and Karin family have been found in this work to be about 10 times smaller than the values of Asphaug (1997) and that of Nesvorný et al. (2006). Results of asteroid-asteroid collisions have attracted special attention in connection with the DART mission. (DART is an acronym for NASA's Double Asteroid Redirection Test. An impactor, on 2022 September 26, hit onto the asteroid Dimorphos, a small companion of the asteroid Didymos, the larger member of the binary asteroid system. DART's impact altered the orbit of Dimorphos about Didymos.) For example, Raducan et al. (2021) studied impacts on asteroids focusing on the expected results of the DART mission. They wrote that ‘numerical studies and laboratory experiments of oblique impacts suggest that crater volume and crater diameter decrease with decreasing impact angle, in a manner that is approximately consistent with the idea that only the vertical component of the impact velocity contributes to the growth of the crater in an oblique impact’. Our results concern the head-on impacts, that are the most efficient for cratering and for disruption. It is worth noticing that the ratio of the radius of the largest crater on the rocky target to the radius of that target is not larger than 0.4, approximately (Leliwa-Kopystynski et al. 2008). Thus, the mass ratio is about 0.064. The line (crater radius) = 0.4R_PB corresponds to the limit separating the cratering regime from the catastrophic disruption regime. These figures have been found considering the largest craters observed on seven asteroids (Juno, Mathilde, Ida, Eros, Gaspra, Annefrank, and Ida's satellite Dactyl) and on two rocky satellites (Phobos and Deimos). All of them have radii not larger than 120 km.

2 PARAMETERS AND DATA INVOLVED INTO THIS WORK

Two parameters (v_cut and C) play a crucial role in choosing the most reliable population of AF. One significant physical property (albedo, p_v) and one observational feature (absolute magnitude, H) are also related to the sizes of individual AFMs. The parameters v_cut, C, and p_v of the LMs are discussed first. The values applied in this work are gathered in columns 3–5 of Table 1. Next, in columns 6a and 6b, we report the population numbers N. The main results of this work are in column 12, which present the lower limit of the size of the impactor that created a particular AF. A detailed description of the content of Table 1 is given below.

Table 1.

The physical data and the parameters as well as the results for 16 AFs considered in this paper.

1 FIN (a)	2 Family name (a)	3 vcut (m s−1) (a)	4 C (10−4 au) (a)	5a Albedo pv (a)	5b Albedo pv (b)	6a N (a)	6b N (this work)	7 RLM (km) (c)	8 RPB (km)	9 ρ (kg m−3)	10 f	11 vimp (km.s−1)	12 rimp,min (m)
401	4Vesta non-dif.	50	1.5 ± 0.5	0.35	0.4228	15252	19170	262.5	262.571	3456 (d)	0.00078	5.089	4227
502	15 Eunomia	50	2.0 ± 0.7	0.19	0.248	5670	11571	115.844	118.048	2834	0.055	5.266	4135
404	20 Massalia	55	0.25 ± 0.05	0.22	0.241	6428	26285	67.84	68.612	3224	0.033	5.089	1517
602	24 Themis	60	2.5 ± 1.0	0.07	0.067	4782	5835	99	128.917	2780	0.55	4.634	10426
606	221 Eos	45	1.5 ± 0.5	0.13	0.166	9789	16805	47.734	61.330	2710	0.53	4.634	2972
517	3815 Konig	55	0.06 ± 0.03	0.04	0.056	354	381	20.643	57.913	2710	0.96	3.670	3523
605	158 Koronis	45	2.0 ± 1.0	0.15	0.213	5949	6879	19.512	20.643	2710	0.16	3.670	396
621	158 Koronis(2)	8	0.01 ± 0.005	0.14	0.213	246	337	10.064	15.084	2710	0.70	3.670	358
610	832 Karin	10	0.003 ± 0.001	0.21	0.221	541	1086	8.5	10.404	2710	0.46	3.670	174
609	490 Veritas	30	0.2 ± 0.1	0.07	0.059	1294	1835	59.402	64.321	3000 (c)	0.21	4.634	2568
406	163 Erigone	50	0.2 ± 0.05	0.06	0.033	1776	2742	40.790	46.075	3000 (c)	0.31	5.089	1544
501	3 Juno	55	0.5 ± 0.2	0.25	0.214	1684	2866	123.298	123.361	3680 (c)	0.0014	5.266	1480
504	128 Nemesis	50	0.25 ± 0.05	0.05	0.067	1302	5835	81.258	82.270	1820 (c)	0.052	5.266	1708
505	145 Adeona	50	0.7 ± 0.3	0.07	0.061	2236	10081	63.892	85.125	1820 (c)	0.58	5.266	4216
403	298 Baptistina	45	0.25 ± 0.05	0.16	0.131	2500	3120	10.568	20.349	2000 (c)	0.86	5.089	443
901	31 Euphrosyne	120	1.0, thiswork. No data in (a)	0.06	0.053	2035	2286	133.54	133.813	1180 (c)	0.0061	4.634	2003

Note. The columns are as follows: 1. Family identification number (FIN), from (a). 2. Family name, from (a). 3. Cut-off velocity v_cut, taken from (a), and next used by us to calculate the values of N listed in column 6b. For FIN 406, we used v_cut = 51 m s⁻¹ instead of v_cut = 50 m s⁻¹ used in (a). 4. Parameter C, from (a). These values were applied by us, however, disregarding the intervals given in (a). 5a, 5b. Albedo pv of the main member of the family. 6a, 6b. Population N of the family. Figures in column 6a are from (a). We obtained figures in column 6b using HCM applied to the data basis containing 463 653 asteroids each time. 7. Radius of the largest member RLM. 8. Radius of the parent body RPB. The parent bodies of all AFs are assumed to be non-differentiated. 9. Density of the LM. It is assumed that the density of an impactor is the same as that of LM. 10. Coefficient f is the fraction of mass lost by the PB during an impact. 11. Mean value of the impact velocity vimp. 12. Lower limit of the radius of the impactor rimp,min, calculated using equation (10). References. (a) Nesvorný et al. (2015). (b) JPL NASA https://ssd.jpl.nasa.gov/tools/sbdb_lookup.html#/(2022). (c) Carry (2012). (d) Russel et al. (2012).

Cut-off velocity. The most widely used distance metric in the proper element domain (a, e, i) [see Zappala et al. (1994) and Masiero et al. (2013) for a discussion on their implementation and possible modifications] between asteroids (1) and (2) is

$$\begin{eqnarray} v\ = {v_{\mathrm{ orb}}}{[{C_a}{(\Delta a/a)^2} + {C_e}{(\Delta e)^2} + \ {C_i}{(\Delta \sin\left( i \right))^2}]^{1/2}}. \end{eqnarray}$$

(1)

Here, Δa = abs(a₁ − a₂), Δe = abs(e₁ − e₂), and Δsin(i) = abs[sin(i₁) − sin(i₂)]. The indexes (1) and (2) denote the two bodies in consideration. The v_orb is the heliocentric velocity of an asteroid on an orbit having the semimajor axis a = (a₁ + a₂)/2. The dimensionless parameters C_a = 5/4, C_e = 2, and C_i = 2 are the weighting factors. Other choices of C_a, C_e, and C _i are also possible (Zappala et al. 1994; Masiero et al. 2013). To make the right choice of the parameter v_cut for each AF considered, we have made a plot of the number n of the potential family members versus v_cut (Wlodarczyk & Leliwa-Kopystynski 2014; Leliwa-Kopystynski & Wlodarczyk 2020). An abrupt (near step-like) increase of the number n following a smooth increase of v_cut indicates leaving the AF area and entering the region of asteroids not associated with this AF. In our calculations, we have used the values of v_cut following Nesvorný et al. (2015).

Parameter C. It is related to a choice of the shape of an AF in (a, H) space, where H is the absolute magnitude of a potential AFM, while a and a_LM are the semimajor axes of that potential AFM and the LM, respectively. Parameter C appears in the so-called V-shape criterion introduced by Vokrouhlidský et al. (2006) and improved by Brož et al. (2013) and Bryce et al. (2018):

$$\begin{eqnarray} H = 5\log\frac{{\left| {a - {a_{\mathrm{ LM}}}} \right|}}{C}. \end{eqnarray}$$

(2)

This formula represents a pair of the ∼V-shaped lines forming a funnel in the (a, H) plane with the vertical axis of symmetry at a = a_LM. Parameter C regulates the steepness of this funnel. Asteroids with a being inside this funnel are treated, as potential family members. In Table 1, column 4, there are quoted original figures for C given by Nesvorný et al. (2015) and applied in this work. For each potential AFM, we used the value of H taken from the data base https://newton.spacedys.com/astdys/index.php?pc=5 (2022). From this first selection of family members, we obtain a range of possible albedo values. Asteroids that have an albedo that differs substantially from the albedo of the LM are rejected as interlopers. The latter are frequently in the upper part of the funnel and thus their contribution to (potential) AF mass is minor. We performed several numerical tests in which the role of parameters C and albedo p_v of potential members were tested.

Albedo p_v of the main member of the family. [Note that the main member does not have to be the same as the LM. The ‘main member’ of AF denotes the asteroid that has given its name to the name of the AF. It does not have to be the largest member of the family. For example: the main member of the Koronis family (FIN_605) is the asteroid 158 Koronis (diameter 35.4 km), whereas the largest member of this family is 208 Lacrimosa (diameter 41.0 km). Diameters are from https://ssd.jpl.nasa.gov/tools/sbdb_lookup.html#/?sstr=158.] The values of albedo for the LMs and the values of their errors (not used in this work) are according to JPL NASA (2022). For any AFM different than LM, its albedo: (i) is either known, which is an exceptional case, or (ii) it is not known (the typical case), and thus its value is assumed to be equal to the albedo of the LM. In Table 1, columns 5a and 5b are given to illustrate differences among the sources. We applied the values from column 5b, disregarding their intervals listed in the source.

Population N of the family. Figures in column 6a are from (a). We obtained the figures in column 6b using HCM applied to the data base containing 463 653 asteroids. Except for the family with FIN 901, our N-values are greater than those listed in source (a). This is obvious since the data base used by us is newer and thus larger than that of Nesvorný et al. (2015). We are aware that the N values in Table 1 are subject to an error resulting from the escape of asteroids from the area (a, e, i) where the AFMs were originally located.

Radius R_LM of the largest member LM. This is the radius of the spherical body having a volume equal to the volume of the LM of a given AF. In Table 1 we listed the values R_LM for all AFs considered in this paper. Except for Vesta, the values of R_LM are known very poorly. The values of radii of Eos and Koronis are very controversial, and thus any result concerning their families presented in the next columns of Table 1 must be considered with strong caution or even treated as very doubtful.

Radius R_PB of the parent body PB. To calculate the radius R_PB the balance of mass is used. The mass M_PB of the PB with the mass m_imp of an impactor is equal to the sum of the mass M_LM of the LM (whose consecutive number n = 1) and the masses of all AFMs with the numbers n ≥ 2:

$$\begin{eqnarray} M_{\mathrm{ PB}}^{} + m_{\mathrm{ imp}}^{} = M_{\mathrm{ LM}}^{} + \mathop \sum \limits_2^N m_n^{} + \sum m_{{\mathrm{ tailmembers}}}^{}. \end{eqnarray}$$

(3)

Here, an asteroid with the number n = N is the smallest observed AFM. The third term on the right-hand side of equation (3) is the sum of the masses of all non-observed family members, that is the mass of a tail. This sum could be estimated using extrapolation of the distribution function n(H) of AFMs for magnitudes beyond the observation limit. This has been studied previously by us (Leliwa-Kopystynski and Wlodarczyk, 2020). Equation (3) can be written for a case when an AF originates from an impact on differentiated PB. In that case, the balance of mass (3) transforms for the balance of volume:

$$\begin{eqnarray} R_{\mathrm{ PB}}^3 &=& R_{\mathrm{ LM}}^3 + \left( {{\rho _{{\mathrm{ mantle}}}}/\rho } \right)\left( {\mathop \sum \limits_2^N r_n^3 + \sum r_{{\mathrm{ tailmembers}}}^3} \right) \\ &&- \left( {{\rho _{\mathrm{ imp}}}/\rho } \right)r_{\mathrm{ imp}}^3. \end{eqnarray}$$

(4)

Here, the mean density ρ is the same for PB and LM. All the fragments (the observed ones as well as those belonging to the unobserved tail) have the same density ρ_mantle equal to the density of mantle material of the differentiated PB; the density of the impactor is ρ_imp. If one assumes that all the densities are the same for all the bodies involved, then equation (4) writes in a simpler form:

$$\begin{eqnarray} R_{\mathrm{ PB}}^3 = R_{\mathrm{ LM}}^3 + \mathop \sum \limits_2^N r_n^3 + \sum r_{{\mathrm{ tailmembers}}}^3 - r_{\mathrm{ imp}}^3. \end{eqnarray}$$

(4’)

Equation (4’) can be applied for all but one AFs considered. The Vesta family is an exception since its PB was the differentiated body. So, for the Vesta family equation (4) is appropriate.

Degree of destruction of the target f (or mass lost coefficient). It is defined as the ratio (M_PB − M_LM)/M_PB, that is the ratio of the mass of all but the first one AFMs to the mass of the PB. Neglecting the mass of the impactor there is:

$$\begin{eqnarray} f = \frac{{\mathop \sum \nolimits_2^N {m_n} + {m_{{\mathrm{ tail}}}}}}{{{M_{\mathrm{ PB}}}}}. \end{eqnarray}$$

(5)

The degree of destruction for the families of Vesta, Eunomia, Massalia, Themis, Konig, Koronis, and Veritas for the cases ‘without the tail’ and ‘with the tail’ were calculated by Leliwa-Kopystynski and Wlodarczyk (2020). Results differ by less than 10 per cent. In this paper, the tails are neglected, and thus formula (5) writes as follows:

$$\begin{eqnarray} f = \frac{{\mathop \sum \nolimits_2^N r_n^3}}{{R_{\mathrm{ LM}}^3 + \mathop \sum \nolimits_2^N r_n^3}} = \frac{{D_{{\mathrm{ frag}}}^3}}{{D_{\mathrm{ LM}}^3 + D_{{\mathrm{ frag}}}^3}}. \end{eqnarray}$$

(6)

The expression in the right-hand side of this formula is given to get compatibility with the notation of Nesvorný et al. (2015). D_LM and D_frag are the diameter of the LM and the diameter of the sphere made of all fragments except the LM, respectively.

Loss of the gravitational energy ΔE_grav of the PB due to loss of a fraction of its mass. The gravitational energy E accumulated in the uniform spherical mass M with radius R is equal to

$$\begin{eqnarray} {E_{{\mathrm{ grav}}}} = \frac{3}{5}G\frac{{{M^2}}}{R} = \frac{{16{\pi ^2}}}{{15}}G{\rho ^2}{R^5}. \end{eqnarray}$$

(7)

Here, G = 6.6743 × 10⁻¹¹ m³ kg⁻¹ s⁻² is the gravity constant. Working with the densities and the radii is more convenient than using the masses and the radii. Applying formula (7) to the PB and the LM, we calculate the difference ΔE_grav between the initial gravitational energy accumulated in the PB and the gravitational energy that remained in the LM after the impact:

$$\begin{eqnarray} \Delta {E_{{\mathrm{ grav}}}} = \frac{{16{\pi ^2}}}{{15}}G{\rho ^2}\left( {R_{\mathrm{ PB}}^5 - R_{\mathrm{ LM}}^5} \right). \end{eqnarray}$$

(8)

This energy represents only a fraction of the impactor energy provided to the target. The rest are used for breaking molecular bindings in PB material and for accelerating fragments to speeds prevailing the escape velocity. In equation (8), the gravitational energy of impactor itself is neglected.

Densities ρ of the LM and AFMs, and ρ_imp of the impactor. In most cases, the density of the LM is very little known, and thus it has to be reasonably adopted. The densities of the majority of the AFMs are not known at all. The figures for the density of PBs presented in column 9 of Table 1 are from our previous work, or they are given by Carry (2012). All values of density are controversial. Of course, nothing is known about the density of an impactor. Therefore, we have to assume them somehow. All that can be said is that the impactor is made of ‘typical’ asteroid material with a density close to the mean density of the asteroids. Let an impactor density ρ_imp is expressed by the mean density of the target ρ_imp = ηρ (for the case when the target is non-differentiated) or by the mantle density ρ_imp = ηρ_m, (for the case when the target is differentiated, the Vesta case). The parameter η has a value close to one. The most convenient (and not worse than another) is an assumption that all the members of a given AF as well as the impactor have the same density as the LM. Such an approach is reasonable, provided that the density of LM is not significantly larger than the mean density of asteroids, as it is for Vesta. Calculations for the size of the impactor on the BP of the Vesta family are given in Section 4.

Impact velocity v_imp. In the asteroid belt, the speeds of collisions between asteroids are a few km/s. More precisely, their values depend on which region of the asteroid belt the impactor comes from and in which region of the belt the impacted body (in our case, the PB of a future AF) is orbiting. Cibulkova et al. (2014) calculated intrinsic collisional probability and mutual impact velocities in the six sub-belts of asteroids. Based on their results, we calculated the weighted averages of mutual impact velocities (Table 2) and applied them in this work. It is worth noting that Zain et al. (2021) applied results of Cibulkova et al. (2014) for their analysis of cratering on Ceres and Vesta.

Table 2.

This table presents the weighted average of mutual collision velocities of asteroids. An impacted asteroid is located in one of the six belts listed in the first column. It gets hit by an asteroid originating from any of the belts listed in the second column. 21 values of the impact velocity and the probabilities of occurring of an impact [calculated by Cibulkova et al. (2014), but not cited here] were used to calculate the weighted average of mutual impact velocities. The average velocities, weighted by the probability of occurrence of a collision, are given in the third column. Note that the names of the belts, as well as their ranges used by Cibulkova et al. (2014), differ slightly from those used by Nesvorný et al. (2015). Note that the largest members of all 16 AFs considered in this paper have semi-axes of the range 2.1–3.28 AU and inclinations less than 17.5°.

Belt	Ranges of major semi-axis in AU (and resonances with Jupiter). The condition for orbital inclination is in the bottom row. From Cibulkova et al. (2014).	Weighted average of mutual impact velocities with asteroids from any of all six belts vimp (m s−1)
Inner belt	2.1 < a < 2.5 (3:1 J)	5089
Middle belt	2.5 < a < 2.823 (5:2 J)	5266
Pristine belt	2.823 < a < 2.956 (7:3 J)	3670
Outer belt	2.956 < a < 3.28 (2:1 J)	4634
Cybele belt	3.28 < a < 3.51	5042
High-inclination region	sin i > 0.34 (i > 20°) (ν6 secular resonance)	8920

Impactor radius, r_imp. During an impact, a fraction of energy (1–ε)E_imp, is accumulated in the LM. It is used to break off the intermolecular bonds inside its matter (deformation, crushing, powdering, and so on). The second fraction, the product εE_imp, diminishes the PB gravitational energy by rejecting a fraction of its mass. Gravitational interaction has a larger role when the size of the PB increases. For a case where the PB mass is larger than the mass of the impactor by orders of magnitudes, the simplified balance of energy is

$$\begin{eqnarray} 0.5\varepsilon {m_{\mathrm{ imp}}}v_{\mathrm{ imp}}^2 = \Delta {E_{{\mathrm{ grav}}}},\,\,\,\,{\rm{where}}\,\,0\ < \varepsilon \le \ 1. \end{eqnarray}$$

(9)

Working with the densities and the radii is more convenient than using the masses and the radii, equations (8) and (9) lead to this formula for the radius of the impactor:

$$\begin{eqnarray} r_{\mathrm{ imp}}^3 = \frac{1}{\varepsilon }\frac{{8\pi }}{5}G\frac{{{\rho ^2}}}{{{\rho _{\mathrm{ imp}}}}}v_{\mathrm{ imp}}^{ - 2}\left( {R_{\mathrm{ PB}}^5 - R_{\mathrm{ LM}}^5} \right). \end{eqnarray}$$

(10)

3 DISCUSSION OF RESULTS: GENERAL

As seen from formula (10), the impactor radius r_imp scales as ε^−1/3. For its numerical value the parameter ε is of the utmost importance. The actual size of the AF-forming impactor ranges from the minimum size r_imp.min corresponding to ε = 1 to this size multiplied by the ε^−1/3. Formula (10) shows that r_imp.min is proportional to ρ^1/3v_imp^−2/3. An estimation of the relative error Δr_imp,min/r_imp,min made using the method of complete differential for the assumed values of the relative error of the impactor density Δρ_imp/ρ_imp = 0.5 (more or less arbitrary), error of the PB density Δρ/ρ = 0.2, and error of the collision velocity Δv_imp/v_imp = 0.1 (according to Tables 1 and 2). These produced a value of the relative error Δr_imp,min/r_imp,min < 0.5. The error of r_imp.min originated from uncertainty of the radii R_PB and R_LM is not discussed here, since it should be considered individually for any family. A rough analysis of the uncertainty of the physical data allows estimating that the minimal values of impactor sizes are determined with accuracy no worse than about –50 per cent to +100 per cent of the values that we calculated and presented in Table 1. The analysis of the error given above is a general one. For any particular AF, an individual approach leading to smaller errors can be implemented, provided that more data is available. This has been done for the Vesta family.

4. DISCUSSION OF RESULTS: VESTA

Among the considered AFs, the Vesta family is unique since we know much more about its LM than what is available for other families. It has been shown by the Dawn mission, that Vesta is a differentiated body with an outer layer less dense than the inner part (the core) with a radius of about 110 km. Vesta mean radius (262.7 km) and mean density (3456 kg m⁻³) are well known (Russell et al. 2012) and their errors can be disregarded. Two large impact-originated basins are identified on the Vesta surface (Jutzi et al. 2013; Zain et al. 2021; Bottke and Jutzi 2022). They are Rheasilvia and Veneneia with diameters of about 500 and 400 km, respectively. Their age is estimated at 3.2–3.5 and ∼1 Ga, respectively (Bottke and Jutzi 2022). A very rough estimate of the relative amount of material ejected from the impact basins scales within an interval from (5/4)² (surface scaling) to (5/4)³ (volume scaling). Here, we adopted the mean value of 1.75. Therefore, the relative share of the collisions that created craters Rheasilvia and Veneneia, and as a consequence, formed the two sub-families is 0.64 and 0.36, respectively. The relative pristine populations of the two sub-families of the Vesta family ought to be within the same limits. It is not possible to identify from which event any family member originates. It is thought that the HED meteorites (Howardites, Eucrites, and Diogenites) have originated from the outer part of the differentiated PB of Vesta. If so, the material of the Vesta outer layer is known from the meteorites that fall on the Earth. Laboratory studies of these meteorites (Macke et al. 2011) showed that their bulk density ranged between 2.62 and 3.37 g cm⁻³. So, an assumption that the density of the Vesta outer layer and thus the density of the Vesta family members (the Vestoids) falls within this range is justified.

We consider two approaches to the internal structure of PB of the Vesta family and thus to the structure of the present Vesta. They are: (A) The non-differentiated pre-Dawn model and (B) the differentiated post-Dawn model that is presently widely accepted.

(A) The simplest but unrealistic approach is considered here, only to compare its results with those obtained by other, more realistic models, presented later on. Equation (10) with the values ρ = 3456 kg m⁻³, R_PB = 2.62571 × 10⁵ m, R_LM = 2.62500 × 10⁵ m (Table 1), and v_imp = 5089 m s⁻¹ (Table 2) gives the radius r_imp = 6.39 × 10⁴ε^−1/3ρ_imp^−1/3 of an impactor that fell on the undifferentiated PB of the Vesta family. Assuming that 1500 kg m⁻³ < ρ_imp < 3000 kg m⁻³ we get an interval of 4430ε^−1/3 m < r_imp,min < 5582ε^−1/3 m. It should be noted that these results do not take into account either the very likely double structure of the Vesta family nor the differentiated structure of its PB.

(B) To perform calculations for a differentiated body, the following formula must be used instead of equation (7):

$$\begin{eqnarray} {E_{{\mathrm{ grav}}}} = \frac{{16{\pi ^2}}}{{15}}G\left[ {\rho _\mathrm{ c}^2R_\mathrm{ c}^5 + \rho _\mathrm{ m}^2\left( {{R^5} - R_\mathrm{ c}^5} \right)} \right]. \end{eqnarray}$$

(11)

It presents gravitational energy accumulated in a spherical body of radius R that is composed of the uniform core with radius R_c and density ρ_c, and the uniform mantle with a thickness (R − R_c) and the density ρ_m. Equation (11) is then used twice. First, for R = R_LM + ΔR, and, secondly, for R = R_LM. Subtracting the second equation from the first one we get a formula for the loss of a part of the gravitational energy of a PB, ΔE_grav = E_{grav, PB} − E_{grav, LM}, after the loss of a part of its mass:

$$\begin{eqnarray} \Delta {E_{\mathrm{grav}}} = \frac{{16{\pi ^2}}}{{15}}G\rho _\mathrm{ m}^2\left[ {{{\left( {{R_{\mathrm{ LM}}} + \Delta R} \right)}^5} - R_{\mathrm{ LM}}^5} \right]. \end{eqnarray}$$

(12)

Here, ΔR = (R_PB − R_LM) is how the radius of an impacted PB decreases due to the loss of mass. Equation (12) differs from equation (8) only by replacing the mean density ρ by the density of mantle material ρ_m. The material of the PB is removed only from its mantle but not from its core. For the case of Vesta, there is ΔR/R_LM << 1 and thus equation (12) takes the form

$$\begin{eqnarray} \Delta {E_{\mathrm{grav}}} = \frac{{16{\pi ^2}}}{3}G\rho _\mathrm{ m}^2R_{\mathrm{ LM}}^4\Delta R. \end{eqnarray}$$

(13)

Introducing this formula to equation (9) we get the radius of the impactor

$$\begin{eqnarray} r_{\mathrm{ imp}}^3 = \frac{1}{\varepsilon }8\pi G\frac{{\rho _\mathrm{ m}^2}}{{{\rho _{\mathrm{ imp}}}}}v_{\mathrm{ imp}}^{ - 2}R_{\mathrm{ LM}}^4\Delta R. \end{eqnarray}$$

(14)

At first, we apply equation (14) assuming that the Vesta family was formed by only one impact event onto a differentiated PB. Putting ρ_imp = ηρ_m, equation (14) writes as follow

$$\begin{eqnarray} r_{\mathrm{ imp}}^3 = \frac{1}{{\varepsilon \eta }}8\pi G{\rho _\mathrm{ m}}v_{\mathrm{ imp}}^{ - 2}R_{\mathrm{ LM}}^4\Delta R. \end{eqnarray}$$

(15)

We put as previously 5089 m s⁻¹, R_LM = 2.625 × 10⁵ m and, following Table 1, we take ΔR = 71 m. Following Macke et al. (2011) we assume that ρ_m = 3000 kg m⁻³. Equation (15) gives the radius of the sole impactor r_imp = 4031(εη)^−1/3 that was able to form the Vesta family trough an impact on the differentiated BP.

Up to now, we performed estimates of the size of the only impactor that was able to create the whole Vesta family at once. Now let's look at the two collisions that formed independently two large impact basins and thus two subfamilies, next perfectly mixed. To do so, equation (13) is written two times, namely for ΔR₁ and for ΔR₂, where the sum (ΔR₁ + ΔR₂) is equal to ΔR, defined previously. The ΔR is divided into two parts in a proportion of 0.64 to 0.36. For ΔR = 71 m there is ΔR₁ = 45.4 m and ΔR₂ = 25.6 m for larger and for smaller events, respectively. With these values of ΔR₁ and ΔR₂, and other parameters equal to their values as previously used, equation (15) leads to the following estimates for the radii of the impactors that formed the double Vesta family: for the impactor that formed the Rheasilvia basin (the larger basin): r_{imp,Rheasilvia} = 3473(εη)^−1/3 m, and for the impactor that formed the Veneneia basin (the smaller basin): r_imp = 2869(εη)^−1/3 m. The role of the parameter η is minor and for a rough estimate it can be assigned to 1. As related to the parameter ε we know only that 0 < ε ≤ 1. Putting ε = 1 we get the minimal values of the radii of impactors r_{imp.min,Rheasilva} = 3473 m and r_{imp,min,Veneneia} = 2869 m. Bottke and Jutzi (2022) state that according to their numerical models the Rheasilvia and Veneneia basins were created by the impact of ∼60–70 km projectiles. These values correlated with our results leading to a conclusion that only the fraction ε ≈ 0.001 of the impact energy was used to form the Vesta family.

5 CONCLUDING REMARKS

We calculated the minimal sizes of impactors that were able to form the AFs of collisional origin. The calculations were based on the considerations concerning the impact velocity and the reasonable assumptions related to densities of the targets and those of the impactors. The data concerning the population of families and, thus, their masses are crucial for calculations. The minimal radii of impactors that were able to form the 15 AFs considered here are listed in Table 1. They are determined with an accuracy of about –50 per cent (down) to +100 per cent (up) for the values that we calculated and presented in Table 1. The efficiency ε of using of the impact energy for the formation of an AF is applied as a parameter throughout the whole paper. In the case of the formation of the double family of Vesta, the ratio of the impactor density to the density of the Vesta mantle, ρ_imp/ρ_m= η, is a second parameter.

ACKNOWLEDGEMENTS

The authors thank the anonymous reviewers much for their critical remarks and carefully improving their English.

DATA AVAILABILITY

The data underlying this paper will be shared on reasonable request to the corresponding author.

APPENDIX

This table presents the highest values of the absolute magnitudes H_max of the numbered asteroids in the close vicinity of the LM of the AF (column 4), with those for the families considered in this work (column 6).

1	2	3	4	5	6
4 Vesta	2.36	17346	19.12	19170	18.7
15 Eunomia	2.64	17521	18.12	11571	18.1
20 Massalia	2.41	15651	19.10	26285	19.1
24 Themis	3.15	14605	17.12	5835	17.6
221 Eos	3.01	9724	17.12	16805	17.5
3815 Koenig	2.57	17899	18.12	381	18.1
158 Koronis	2.87	5941	17.12	6879	17.7
832 Karin	2.87	5941	17.12	1086	17.4
490 Veritas	3.17	14805	17.12	1835	17.1
163 Erigone	2.37	17476	19.07	2742	19.1
3 Juno	2.67	5766	18.12	2866	17.9
128 Nemesis	2.75	14146	18.11	5835	18.1
145 Adeona	2.67	15766	18.12	10081	18.1
298 Baptistina	2.26	10908	19.11	3120	18.5
31 Euphrosyne	3.16	15073	17.12	2286	16.4

Note. The columns are as follows: 1. The asteroid that is the LM of the considered AF. 2. The semi-axis of the LM. 3. The population of numbered asteroids within the belt with the thickness (a ± 0.01) au, ref. (a). 4. The highest value of absolute magnitude H_max of a numbered asteroid within the belt (a ± 0.01) au. Ref. (a). 5. The population N of the AF. See Table 1 of this work. 6. The highest value of absolute magnitude H_max of a numbered asteroid within the AF. This work. Reference. (a) JPL NASA https://ssd.jpl.nasa.gov/tools/sbdb_lookup.html#/?sstr=20 as of 2024 January 17.