---
abstract: 'The Hoffmeister family is a C-type group located in the central main belt.  Dynamically, it is important because of its interaction with the ${\nu}_{1C}$ nodal secular resonance with Ceres, that significantly increases the dispersion in inclination of family members at lower semi-major axis.  As an effect, the distribution of inclination values of the Hoffmeister family at semi-major axis lower than its center is significantly leptokurtic, and this can be used to set constraints on the terminal ejection velocity field of the family at the time it was produced.  By performing an analysis of the time behaviour of the kurtosis of the $v_W$ component of the ejection velocity field (${\gamma}_2(v_W)$), as obtained from Gauss’ equations, for different fictitious Hoffmeister families with different values of the ejection velocity field, we were able to exclude that the Hoffmeister family should be older than 335 Myr.  Constraints from the currently observed inclination distribution of the Hoffmeister family suggest that its terminal ejection velocity parameter $V_{EJ}$ should be lower than 25 m/s. Results of a Yarko-YORP Monte Carlo method to family dating, combined with other constraints from inclinations and ${\gamma}_2(v_W)$, indicate that the Hoffmeister family should be $220^{+60}_{-40}$ Myr old, with an ejection parameter $V_{EJ}= 20\pm5$ m/s.'
author:
- |
    V. Carruba$^{1}$[^1], B.Novaković$^{2}$, S. Aljbaae$^{1}$\
    $^{1}$UNESP, Univ.  Estadual Paulista, Grupo de dinâmica Orbital e Planetologia, Guaratinguetá, SP, 12516-410, Brazil.\
    $^{2}$University of Belgrade, Department of Astronomy, Faculty of Mathematics, 11000 Belgrade, Serbia.\
date: 'Accepted 2016 November 18.  Received 2016 November 17; in original form 2016 September 21.' 
title: The Hoffmeister asteroid family
---

\[firstpage\]

Minor planets, asteroids: general – minor planets, asteroids: individual: Hoffmeister –celestial mechanics. 

Introduction {#sec: intro}
============

The Hoffmeister family was identified in @Milani_2014, and more recently in @Nesvorny_2015 with a Family Identification Number (FIN) equal to 519.  As originally observed by Novaković et al. ( 2015), this family is characterized by its interaction with the ${\nu}_{1C}$ nodal secular resonance with Ceres, whose effect is to significantly spread the distribution in inclination of family members for semi-major axis lower than $\simeq 2.78$ au.  @Carruba_2016 also identified this family as one of the eight groups characterized by having the most leptokurtic distribution of the $v_W$ component of terminal ejection velocities field, that is closely related to the inclination through the third Gauss equation (see, for instance, Eq.  3 in @Carruba_2016).  While most families are formed with an originally leptokurtic distribution of ejection velocities (see @Carruba_2016 for an explanation of the role that escape velocities from parent body have in creating originally leptokurtic distributions of terminal ejection velocities), in the absence of dynamical mechanisms able to change the inclination of family members, the distribution of $v_W$ tends in time to become more mesokurtic, or Gaussian.  This, however, is not the case for the Hoffmeister family, whose interaction with the ${\nu}_{1C}$ nodal secular resonance significantly shaped its inclination distribution. 

The peculiar nature of the Hoffmeister family allows for the use of techniques of family dating not available for other asteroid groups.  In particular, the study of the time behavior of the kurtosis of the $v_W$ component of the ejection velocity field (${\gamma}_2(v_W)$), as performed by @Carruba_2016b [@Carruba_2016c] for the Astrid, Gallia, Barcelona, and Hansa families, could provide invaluable constraints on the family age and on the $V_{EJ}$ parameter describing the standard deviation of the initial ejection velocity field, assumed as Gaussian.  By observing by what time the current value of ${\gamma}_2(v_W)$ is reached, for fictitious Hoffmeister families with different values of $V_{EJ}$, and by imposing constraints on the most likely value of this parameter, based on the current inclination distribution of the part of the Hoffmeister family not affected by the ${\nu}_{1C}$ secular resonance, we can then obtain constraints on both the most likely values of family age and $V_{EJ}$, not available for regular families. 

Family identification and dynamical properties {#sec: fam_ide}
==============================================

As a first step in our analysis, we selected the (1726) Hoffmeister family[^2], as identified in @Nesvorny_2015 using the Hierarchical Clustering Method (HCM, [@Bendjoya_2002]) and a cutoff of 45 m/s. 1819 members of the Hoffmeister dynamical group were identified in that work.  @Milani_2014 identified 1905 members of the Hoffmeister family, with an $(a,e)$ and $(a,\sin{(i)})$ distribution very similar to that observed for the @Nesvorny_2015 group.  For the sake of consistency, in this work we will use the data from @Nesvorny_2015, but we will discuss any significant difference with results from @Milani_2014, when appropriate.  Using the criteria defined in @Carruba_2016 we defined as objects in the local background of the Hoffmeister family as those with synthetic proper elements[^3], whose values of proper $e$ and $\sin{(i)}$ are in a range from the family barycenter to within four standard deviations of the observed distribution for the Hoffmeister family, namely from 0.0300 to 0.071 in proper $e$ and from 0.054 to 0.102 in $\sin{(i)}$. The minimum value of proper $a$ was chosen from the minimum value of Hoffmeister members minus 0.02 au, the averaged expected orbital mobility caused by close encounters with massive asteroids over 4 Byr [@Carruba_2013].  The maximum $a$ value was taken at the center of the 5J:-2A mean-motion resonance.  Namely, this corresponds to an interval in $a$ from 2.730 au to 2.825 au.  Overall, we found 4781 asteroids in the local background of the Hoffmeister family so defined, also including known members of other families. 

@Carruba_2009, in his analysis of the (363) Padua family found that the main asteroid families in the local background of the Hoffmeister cluster are the (847) Agnia and Padua groups, both characterized by their interaction with the $z_1$ secular resonance [@Vokrouhlicky_2006b; @Carruba_2009].  This analysis was essentially confirmed by @Nesvorny_2015.  Apart from the cited Agnia and Padua families, these authors also identified the (2732) Witt family at higher inclinations.  Other families in the region, such as the groups of (128) Nemesis, (1668) Hanna, and (1222) Tina, do not have family members in the local background of Hoffmeister defined according to our criteria.  Concerning the two major other families in the region, Padua and Agnia, despite some difference in terms of family membership between the Nesvorný and Milani groups for the Padua family, that has 1087 and 864 members in these two classifications respectively, the corresponding distributions in the $(a, e)$ and $(a, \sin{(i)})$ planes are similar.  The only significant difference in the distribution in the $(a,\sin{(i)})$ plane for the two Padua families is that the Nesvorný group has a small population of objects at $a > 2.78$ au not visible in the Milani group.  The case of the Agnia family is a bit more problematic.  This family has 3033 members in Milani et al. ( 2014), and only 2125 members in Nesvorný et al. ( 2015).  Most of the difference in family memberships is caused by the presence of multi-opposition objects in the Milani group, that are not accounted for in the Nesvorný family.  Also, the Milani Agnia family extends beyond the 3-1-1 three-body resonance, while the Nesvorný one does not.  Quite interestingly, @Milani_2014 and @Spoto_2015 identify the (3395) Jitka sub-family inside the Agnia family, that is quite distinguished in term of physical properties from the rest of the Agnia family, and that will be further discussed in the next section. 

These differences in family memberships for the Agnia and, in a lesser measure, the Padua groups can a bit affect the determination of the local background, but should not change our overall conclusions.  Therefore, since there is a good agreement between the two Hoffmeister families, in terms of family membership and other properties, in this work we will use the families as determined in @Nesvorny_2015. 

![ image](FIGURES/back_ae.eps){width="3.1in"}

![ image](FIGURES/back_ai.eps){width="3.1in"}

After removing members of the Padua, Agnia, and Witt groups, the local background consisted of 1721 objects.  Fig.  \[fig: Back\_hoff\] display a projection as black dots of these asteroids in the $(a,e)$ (panel A) and $(a,\sin{(i)})$ plane.  Members of the Hoffmeister, Padua, Agnia, and Witt families are shown as full blue, green, yellow, and magenta dots.  As originally observed in @Novakovic_2015, the left side of the Hoffmeister family is quite more spread in inclination than its right side, because of the interaction of this family with the ${\nu}_{1C}=s-s_C$ linear secular resonance with Ceres.  To start assessing the dynamical importance of this resonance, we obtained a dynamical map of synthetic proper elements, with the method discussed in @Carruba_2010, for particles subjected under the gravitational influence of all planets, plus Ceres, Pallas, and Vesta[^4].  We integrated 3000 particles in a 50 by 60 grid in osculating initial $(a,\sin{(i)})$ plane, with a step of 0.02 au in $a$ and 0.1 degrees in $i$. Initial values of $a$ and $i$ were 2.73 au and 0 degrees, respectively.  The eccentricity and the other angles of the test particles were those of (1726) Hoffmeister at J2000. 

![ image](FIGURES/Hoffmeister_map_ai.eps){width="3.1in"}

![ image](FIGURES/res_ast.eps){width="3.1in"}

Our results are shown in Fig.  \[fig: map\_ai\_cpv\], panel A. Values of the proper $(a,\sin{(i)})$ of each particle are shown as black dots.  Blue and red full circles are associated with likely resonators of the ${\nu}_{1C}$ and $z_1$ secular resonance, defined as objects whose $s$ and $g+s$ frequencies are to within $\pm 0.2$ arcsec/yr from the proper node frequency of Ceres for the ${\nu}_{1C} =s -s_C$ resonance ($s_c = -59.17$ arcsec/yr, [@Knezevic_2003]), and from the sum of the proper pericenter and node frequencies of Saturn for the $z_1 = g+s-g_6-s_6$ resonance ($g_6+s_6 = 1.90$ arcsec/yr [@Knezevic_2003]).  As discussed in @Carruba_2009, the $z_1$ resonance plays a major role in the dynamical evolution of the Padua family, while the ${\nu}_{1C}$ has a pivotal role in the evolution of the Hoffmeister group [@Novakovic_2015].  To check for the effects of these two resonance on the dynamical evolution of the Hoffmeister family, we also integrated 1819 members of the family and checked the time behavior of the ${\nu}_{1C}$ and $z_1$ resonant arguments over 20 Myr.  We found 180 and 54 members of the family in librating states of these two resonances, respectively.  Their $(a,\sin{(i)})$ projection is shown in Fig.  \[fig: map\_ai\_cpv\], panel B. Hoffmeister members are dispersed in inclination after interacting with the ${\nu}_{1C}$ resonance, as originally observed by @Novakovic_2015.  A significant fraction of Hoffmeister members (about 3%) reached values of inclination large enough to allow for capture into the $z_1$ resonance.  We expect that some of the past members of the Hoffmeister family could have been drawn to the region of the Padua family and be current interlopers of that group. 

Physical properties {#sec: Taxonomy}
===================

After revising the effect of the local dynamics, we now turn our attention to the taxonomic properties of local objects.  @Carruba_2009 discuss properties of asteroids in the orbital proximity of the Padua family, interested readers could find more information in that paper.  Concerning objects in the background of the Hoffmeister family, as defined in this section, we found 287 objects with photometric data in the Sloan Digital Sky Survey-Moving Object Catalog data, fourth release (SDSS-MOC4 hereafter, [@Ivezic_2001]) in thi s region, 71 of which are members of the Hoffmeister dynamical family.  If we consider all available data, regardless of its error, 1515 objects have geometric albedo and absolute magnitude information in the WISE and NEOWISE database [@Masiero_2012].  Fig.  \[fig: Backround\_SDSS\_WISE\] displays asteroids with their classification obtain with the method of @DeMeo_2013 (panel A).  Panel B displays objects with WISE geometric albedo $p_V$ with values compatible with a C-complex taxonomy ($p_V < 0.12$, blue full dots) and with an S-complex taxonomy, ($0.12 < p_V < 0.30$, red full dots, @Masiero_2012.  As can be seen from the figure, the Hoffmeister family is compatible with a C-type composition, the Padua family was probably originated from the break-up of an X-type asteroid, and most of the members of the Agnia family are S-type, which also confirms the analysis of @Nesvorny_2015.  As discussed in @Spoto_2015, the Jitka sub-family in the Agnia family is characterized by higher albedo values than that of the rest of the family.  While most of Agnia members have $p_V = 0.15\pm 0.01$, members of the Jitka sub-family have $p_V = 0.31\pm 0.04$. The Witt family, not visible in the figure, is a S-type group.  The mean albedo value for the Hoffmeister, Padua, and Agnia family is of $(0.05 \pm 0.025), (0.07\pm0.03)$, and $(0.15\pm0.01)$, respectively. 

![ image](FIGURES/background_sdss.eps){width="3.1in"}

![ image](FIGURES/background_pv.eps){width="3.1in"}

Concerning halo objects, in the local background of the Hoffmeister family only the Hoffmeister family itself is associated with a C-type composition.  It is reasonable therefore to conclude that the 95 asteroids with a SDSS-MOC4 C-type compatible composition in the region (see Fig.  \[fig: Backround\_SDSS\_WISE\] could all be potentially associated with the Hoffmeister family.  C-type objects in the region of the Padua family could be potential former members of the Hoffmeister family that reached this region with the mechanism of dynamical diffusion (capture in the ${\nu}_{1C}$ resonance and then in the $z_1$) discussed in the previous section.  If we consider the albedo data, Fig.  \[fig: Wise\_halo\], panel A, displays an histogram of the WISE albedo data for the asteroids members of the Hoffmeister (blue line), Padua (green line) and Agnia (red line) families.  Vertical red dashed lines show the 1-sigma range of albedo values for the Hoffmeister family.  There is a region of overlapping for asteroids belonging to the Hoffmeister and Padua family, (at 1-sigma level the Padua family covers the range of $p_V = 0.07\pm0.03)$, so that it is not possible to distinguish objects originating from one of these two families only based on albedo (the contribution from the Agnia family in this 1-sigma interval is essentially negligible).  If we limit our data to asteroid in the 1-sigma $p_V$ interval ($0.025 < p_V < 0.075$, Fig.  \[fig: Wise\_halo\], panel B), then we can safely conclude that objects at inclination lower than that of the Hoffmeister barycenter should be most likely originate from this family.  However, no conclusion could be safely reached for objects at higher inclinations. 

![ image](FIGURES/pv_hist.eps){width="3.1in"}

![ image](FIGURES/ai_halo.eps){width="3.1in"}

Constraints on terminal ejection velocities from the current inclination distribution {#sec: inc_constr}
=====================================================================================

@Nesvorny_2015 estimate the age of the Hoffmeister family to be $210\pm10$ Myr, while @Spoto_2015, using a V-shape criteria, evaluate the family to be $330\pm100$ Myr old, and found no asymmetry between the left and right side of the family semi-major axis distribution, concerning the estimation of the group age.  As discussed in @Carruba_2016b Monte Carlo methods [@Vokrouhlicky_2006a; @Vokrouhlicky_2006b; @Vokrouhlicky_2006c] that simulates the evolution of the family caused by the Yarkovsky and YORP effects, where YORP stands for Yarkovsky-O’Keefe-Radzievskii-Paddack effect, could also be used to obtain estimates of the age and terminal ejection velocities of the family members (these models will be referred as “Yarko-Yorp” models hereafter). 

However, the age estimates from these methods depend on key parameters describing the strength of the Yarkovsky force, such as the thermal conductivity $K$ and bulk and surface density ${\rho}_{bulk}$ and ${\rho}_{surf}$, that are in many cases poorly known.  Before attempting our own estimate of the family age and terminal ejection velocity field, here we analyze what constraints could be obtained on the possible values of terminal ejection velocities of the original Hoffmeister family from its current inclination distribution. 

Assuming, in first approximation, that the original ejection velocity field of the Hoffmeister family could be approximated as isotropic (see @Carruba_2016 for a discussion of the caveats on this hypothesis), we can model the distribution of ejection velocities of asteroids with a Gaussian distribution whose standard deviation follows the relationship:

$$V_{SD}=V_{EJ}\cdot(5km/D),
\label{eq: V_EJ}$$

where $V_{EJ}$ is the terminal ejection velocity parameter to be estimated, and $D$ is the asteroid diameter.  @Broz_2013 estimated that the ratio of the radius of the parent body with that of the largest fragment was of 0.14, which yields an estimate of 90.85 km for the parent body diameter, and an escape velocity of 39.6 m/s, assuming a mean density for the parent body of 1300 kg/m$^3$, typical of C-type asteroids.  If we only consider objects with $a > 2.795$ au, so as to eliminate the asteroids that interacted with the $s-s_C$ resonance, then the currently observed minimum and maximum values of $\sin{(i)}$ of family members are 0.0742 and 0.0782, respectively.  Neglecting possible changes in $\sin{(i)}$ after the family formation, which is motivated by the fact that the local dynamics does not seems to particularly affect asteroids in this region (see Fig.  \[fig: map\_ai\_cpv\]), these values set constraints on the possible terminal ejection velocity parameter $V_{EJ}$ with which the family was created.  We generated synthetic families for values of $V_{EJ}$ from 5 m/s up to 40 m/s. Fig.  \[fig: inc\_constr\] show an $(a,\sin{(i)})$ projection of the initial orbital dispersion of the members of the family generated for $V_{EJ} = 15$ m/s (panel A) and $V_{EJ} = 25$ m/s.

![ image](FIGURES/inc_constr_15.eps){width="3.1in"}

![ image](FIGURES/inc_constr_25.eps){width="3.1in"}

For $V_{EJ} = 15$ m/s 6 particles (0.3% of the total) had values of $\sin{(i)}$ outside the range of values currently observed and $a > 2.995$ au, while for $V_{EJ} = 25$ m/s these number was 48 (2.7% of the total).  Based on these considerations, it seems unlikely that the ejection velocity parameter $V_{EJ}$ was larger than $25$ m/s, or a larger number of asteroids outside the Hoffmeister family at $a > 2.795$ au would be visible today. 

Ejection velocities evolution {#sec: term_vel}
=============================

As previously discussed, the Hoffmeister family is one of the seven families identified in @Carruba_2016 as being characterized by having a very leptokurtic distribution in the orthogonal component $v_W$ of the currently estimated terminal ejection velocities, and, therefore, of the asteroid inclinations.  As observed in Sect.  \[sec: fam\_ide\], this is mainly caused by the interaction of this family with the ${\nu}_{1C}$ secular resonance that strongly affects the distribution in proper inclination of family members at lower values of semi-major axis, and, therefore, their $v_W$ values.  Recently, @Carruba_2016b investigated how the time evolution of $v_W$ values could be used to set constraints on the initial values of the $V_{EJ}$ parameter for the Astrid family Using the same approach, here we simulated fictitious Hoffmeister families with the currently observed size-frequency distribution, values of the parameters affecting the strength of the Yarkovsky force typical of C-type asteroids according to @Broz_2013, i.e., bulk and surface density equal to ${\rho}_{bulk}={\rho}_{surf} = 1300$ kg/m$^3$, thermal conductivity $K =0.01$  W/m/K, thermal capacity equal to $C_{th} = 680$ J/kg/K, Bond albedo $A_{Bond} =0.02$ and infrared emissivity $\epsilon = 0.9$. The fictitious families had values of $V_{EJ} = 10$, and $20$ m/s, the most likely values of this parameter, according to the analysis of the previous section (values of $V_{EJ}$ larger or equal to 25 m/s were deemed to be incompatible with the current inclination distribution of the Hoffmeister family at large $a$).  Particles were integrated with $SWIFT\_RMVSY$, the symplectic integrator developed by @Broz_1999 that simulates the diurnal and seasonal versions of the Yarkovsky effect, over 400 Myr and the gravitational influence of all major planets plus Ceres.  Values of $v_W$ were then obtained by inverting the third Gauss equation [@Murray_1999]:

$$\delta i = \frac{(1-e^2)^{1/2}}{na} \frac{cos(\omega+f)}{1+e cos(f)} \delta v_W. 
\label{eq: gauss_3}$$

where $\delta i= i-i_{ref}$, with $i_{ref}$ the inclination of the barycenter of the family, and $f$ and $\omega+f$ assumed equal to 30$^{\circ}$ and 50.5$^{\circ}$, respectively.  Results from @Carruba_2016 show that the shape of the $v_W$ distribution is generally not strongly dependent on the values of $f$ and $\omega+f$, except for values of $\omega+f$ close to $\pm 90^{\circ}$ (which does not seems to be the case for the Hoffmeister family, since these values would have produced a very small inclination distribution). 

![ image](FIGURES/VW10.eps){width="2.8in"}

![ image](FIGURES/VW20.eps){width="2.8in"}

Our results are displayed in Fig.  \[fig: gamma2\] for members of a fictitious family with $V_{EJ} = 10$ (Panel A) and $20$ m/s (panel B).  Vertical lines display the maximum range of uncertainty for the family age, according to @Spoto_2015, while the horizontal line shows current value of ${\gamma}_2(v_W) = 2.21$ for the @Nesvorny_2015 Hoffmeister family (the ${\gamma}_2(v_W)$ for the @Milani_2014 family is quite similar and equal to 2.27).  In our computation of ${\gamma}_2(v_W)$ we neglected particles that reached values of $\sin{(i)}$ beyond $\pm 4 \sigma({\sin{(i)}})$ from the family center.  Individual spikes in the time behaviour of ${\gamma}_2(v_W)$ in Fig.  \[fig: gamma2\] are associated with single particles that temporarily approached such extreme values of inclination.  The present value of ${\gamma}_2(v_W)$ is first attained after 320 Myr for the simulation with $V_{EJ} = 10$ m/s and after 260 Myr for the second simulation.  It is lastly attained at 335 Myr in the first simulation and at 280 Myr in the second, if we not consider fluctuations associated with isolated spikes.  Overall, we have an upper limit for the Hoffmeister age of 335 Myr for $V_{EJ} = 10$ m/s and of 280 Myr for $V_{EJ} = 20$ m/s. In the next section we will try to further refine our family age estimate. 

Chronology {#sec: chron}
==========

Now that the analysis of the current inclination distribution and our ${\gamma}_2$ test provided independent constraint on the values of the $V_{EJ}$ parameter, we can try to obtain an independent age estimate for this family.  We use the approach described in @Carruba_2015a that employs a Monte Carlo method [@Vokrouhlicky_2006a; @Vokrouhlicky_2006b; @Vokrouhlicky_2006c] to estimate the age and terminal ejection velocities of the family members.  More details on the method can be found in @Carruba_2015a.  Essentially, the semi-major axis distribution of simulated asteroid families is evolved under the influence of the Yarkovsky effect (both diurnal and seasonal version), the stochastic YORP force, and changes in values of the past solar luminosity.  Distributions of a $C$-target function are then obtained through the equation:

$$0.2H=log_{10}(\Delta a/C),
\label{eq: target_funct_C}$$

where $H$ is the asteroid absolute magnitude, and $\Delta a = a -a_{center}$ is the distance of each asteroid from its family center, here defined as the family center of mass.  For the Hoffmeister family this is essentially equal to the semi-major axis of 1726 Hoffmeister itself.  We can then compare the simulated $C$-distributions to the observed one by finding the minimum of a ${\chi}^2$-like function:

$${\psi}_{\Delta C}=\sum_{\Delta C}\frac{[N(C)-N_{obs}(C)]^2}{N_{obs}(C)},
\label{eq: psi}$$

where $N(C)$ is the number of simulated objects in the $i-th$ $C$ interval, and $N_{obs}(C)$ is the observed number in the same interval.  Good values of the ${\psi}_{\Delta C}$ function are close to the number of the degrees of freedom of the ${\chi}^2$-like variable.  This is given by the number of intervals in the $C$ minus the number of parameters estimated from the distribution (in our case, the family age and $V_{EJ}$ parameter).  Using only intervals with more than 10 asteroids, to avoid the problems associated with small divisors in Eq.  \[eq: psi\], we have in our case 14 intervals for $C > 0$ (see Fig.  \[fig: Nobs\_C\_hoff\], panel A, results from the @Milani_2014 Hoffmeister family are compatible to within the errors) and 2 estimated parameters, and, therefore, 12 degrees of freedom.  We only use positive values of $C$ so as to concentrate on the part of the Hoffmeister family at larger semi-major axis, not affected by the interaction with the ${\nu}_{1C}$ secular resonance.  If we assume that the ${\psi}_{\Delta C}$ probability distribution follows a law given by an incomplete gamma function of arguments ${\psi}_{\Delta C}$ and the number of degrees of freedom, the value of ${\psi}_{\Delta C}$ associated with a 3-sigma probability (or 99.7%) of the simulated and real distributions being compatible is equal ${\psi}_{\Delta C}=2.73$ [@Press_2001]. 

![ image](FIGURES/Nobs_C_hoff.eps){width="3.in"}

![ image](FIGURES/V_AGE_POS.eps){width="3.in"}

Results of our Monte Carlo simulation are shown in Fig.  \[fig: Nobs\_C\_hoff\], panel B. If we neglect values of $V_{EJ} > 25$ m/s, that are not likely to have occurred based on the current inclination distribution of the part of the Hoffmeister family unaffected by the ${\nu}_{1C}$ secular resonance, then our best-fit solution suggests an age of $220^{+70}_{-40}$ Myr and an ejection parameter of $V_{EJ}= 20\pm5$ m/s. Combining these results with those of Sect.  \[sec: term\_vel\] (i.e., an age of less than 280 Myr for $V_{EJ}= 20$ m/s), we can conclude that the most likely age and ejection velocity parameter of the Hoffmeister family are $220^{+60}_{-40}$ Myr and $V_{EJ}= 20\pm5$ m/s.

Conclusions {#sec: conc}
===========

Our results could be summarized as follows:

-   We first identified the Hoffmeister family in the domain of proper element [@Nesvorny_2015], and used the results to define an orbital background region of this group.  Other families in the region are the Padua, Agnia, and Witt groups.  Of the 1819 members of the Hoffmeister dynamical group, 180 asteroids (9.9% of the total) are in librating states of the ${\nu}_{1C}$ secular resonance, and while 54 members (3.0% of the total) are in librating states of the $z_1 =g-g_6+s-s_6$. Most of the Hoffmeister $z_1$ librators are objects at higher inclinations than those typical for the rest of the family, as also confirmed by our results for dynamical maps in the region (see Fig.  \[fig: map\_ai\_cpv\], panel A). 

-   We revised the taxonomic and physical properties of the Hoffmeister family.  The Hoffmeister family is compatible with the break-up of a C-type object of low albedo (the mean geometric albedo of the family is 0.05).  Objects of low albedo ($0.025 < p_V < 0.075$) at lower inclinations than those of the Hoffmeister family may be former family members.  No positive conclusion can be reached for low-$p_V$ objects at higher inclinations, because of possible contamination from the near X-type Padua family. 

-   We computed the fraction of particles that would reach regions in sine of inclination incompatible with the current distribution of Hoffmeister members that are not affected by the ${\nu}_{1C}$ secular resonance, for fictitious families with different values of the $V_{EJ}$ ejection velocity parameter describing the standard deviation of their ejection velocity field, assumed to be Gaussian.  The current inclination distribution of the Hoffmeister family suggests that $V_{EJ}$ should be less than 25 m/s, or a larger fraction of its members would be currently observed at higher and lower inclinations. 

-   We studied the dynamical evolution of two fictitious Hoffmeister families under the influence of the Yarkovsky effect for two values of the $V_{EJ}$ ejection velocity parameter, 10 m/s and 20 m/s. Contrary to the case of the Astrid family, values of thermal conductivity ($K =0.01 W/m/K$) and mean density (${\rho}_{bulk} = {\rho}_{surface}= 1300~kg/m^3$) appropriate for a C-type class family such as Hoffmeister seems to produce results of ${\gamma}_2(v_W)$ evolution compatible with current estimates of the Hoffmeister family age.  Current values of the ${\gamma}_2(v_W)$ parameter for the Hoffmeister family are reached until 335 Myr for $V_{EJ}= 10$ m/s and until 280 Myr for $V_{EJ}= 20$ m/s, which sets upper limits on the Hoffmeister family age. 

-   We compute the age of the Hoffmeister family using a Monte Carlo approach for its Yarkovsky and YORP evolution.  Our best-fit solution, also accounting for the results from the ${\gamma}_2(v_W)$ time behaviour analysis, suggests that the Hoffmeister family should be $220^{+60}_{-40}$ Myr old, with an ejection parameter $V_{EJ}= 20\pm5$ m/s.

Overall, an analysis of the ${\gamma}_2(v_W)$ time behaviour provided invaluable constraints on the age and ejection velocity field of a $v_W$ leptokurtic family, such as Hoffmeister, showing once again, in our opinion, the importance that constraints from secular dynamics (in this case the interaction of the Hoffmeister family with the ${\nu}_{1C}$ secular resonance) could provide in asteroid dynamics. 

Acknowledgments {#acknowledgments .unnumbered}
===============

We are grateful to the reviewer of this paper, Dr. Federica Spoto, for comments and suggestions that greatly improved the quality of this work.  We would also like to thank the São Paulo State Science Foundation (FAPESP) that supported this work via the grant 16/04476-8, and the Brazilian National Research Council (CNPq, grant 305453/2011-4).  BN acknowledges support by the Ministry of Education, Science and Technological Development of the Republic of Serbia, grant 176011.  This publication makes use of data products from the Wide-field Infrared Survey Explorer (WISE) and NEOWISE, which are a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. 

Bendjoya, P., Zappal[à]{}, V. 2002.  Asteroids III, W. F. Bottke Jr., A. Cellino, P. Paolicchi, and R. P. Binzel (eds), University of Arizona Press, Tucson, 613. 

Brož, M., 1999.  Thesis, Charles Univ.,  Prague, Czech Republic. 

Brož, M., Morbidelli, A., Bottke, W. F., Rozehnal, J., Vokrouhlický, D., Nesvorný, D., 2013, A&A, 551, A117. 

Carruba, V. 2009.  MNRAS, 395, 358. 

Carruba, V. 2010.  MNRAS, 408, 580. 

V. Carruba, M. Huaman, R. C. Domingos, F. Roig 2013.  A&A 550, A85. 

Carruba, V., Nesvorný, D., Aljbaae, S., Domingos, R. C., Huaman, M. E., 2015a, MNRAS, 451, 4763. 

Carruba, V., Nesvorný, D., 2016, MNRAS, 457, 1332. 

Carruba, V. 2016.  MNRAS, 461, 1605

Carruba, V. Nesvorný, D., Domingos, R. C., Aljbaae, S., Huaman, M., 2016, MNRAS, 463, 705. 

DeMeo, F. E., Carry, B., 2013, Icarus, 226, 723. 

Ivezić, Ž, and 34 co-authors, 2001, AJ, 122, 2749. 

Knežević, Z., Milani, A., 2003, A&A, 403, 1165. 

Masiero, J. R., Mainzer, A. K., Grav, T., Bauer, J. M., and Jedicke, R., 2012, APJ, 759, 14. 

Milani, A., Cellino, A., Kne[ž]{}evi[ć]{}, Z., et al.  2014, Icarus, 239, 46. 

Murray, C. D., Dermott, S. F., 1999, Solar System Dynamics, Cambridge Univ.  Press, Cambridge. 

Nesvorný, D., Brož, M., Carruba, V. 2015, In Asteroid IV, (P. Michel, F. E. DeMeo, W. Bottke Eds.),  Univ.  Arizona Press and LPI, 297. 

Novakovi[ć]{}, B., Maurel, C., Tsirvoulis, G., & Kne[ž]{}evi[ć]{}, Z. 2015, , 807, L5

Press, V.H., Teukolsky, S. A., Vetterlink, W. T., Flannery, B. P., 2001, Numerical Recipes in Fortran 77, Cambridge Univ.  Press, Cambridge. 

Spoto, F., Milani, A. Knežević, Z. 2015, Icarus, 257, 275. 

Vokrouhlick[ý]{}, D., Bro[ž]{}, M., Morbidelli, A., et al.  2006a, Icarus, 182, 92. 

Vokrouhlický D., Brož, M., Bottke, W. F., Nesvorný, D., Morbidelli, A. 2006b, Icarus, 183, 349. 

Vokrouhlický D., Brož, M., Bottke, W. F., Nesvorný, D., Morbidelli, A. 2006c, Icarus, 183, 349. 

\[lastpage\]

[^1]: E-mail: vcarruba@feg.unesp.br

[^2]: For the sake of brevity, the identification number of families discussed in this paper will only be provided once.  After that, families will be referred to only by the parent body name. 

[^3]: We use data from the AstDyS site (http://hamilton.dm.unipi.it/astdys, @Knezevic_2003), accessed on July 3$^{rd}$, 2016. 

[^4]: Dynamical maps in the $(a,e)$ and $(a,\sin{(i)})$ for the cases without massive asteroids were obtained for the region of the Padua family in @Carruba_2009.  Interested readers could find more information about these results in that paper.