Ingela Alger
email: ingela.alger@tse-fr.eu

Jean-François Laslier
email: jean-francois.laslier@ens.fr

Homo moralis goes to the voting booth: a new theory of voter turnout *

Keywords: voter turnout, voting, ethical voter, homo moralis, Kantian morality

Why do voters incur costs to participate in large elections? This paper proposes an exploratory analysis of the implications of evolutionary Kantian morality for this classical problem in the economic theory of voting: the costly participation problem.

Introduction

While large democratic elections are a means for voters to influence the outcome, the rational theory literature [START_REF] Downs | An Economic Theory of Democracy[END_REF]) has shown that purely consequentialist motives cannot explain the turnout rates observed in large political elections. This "paradox of voting" is due to the low probability for each voter of being pivotal [START_REF] Ledyard | The pure theory of large two-candidate elections[END_REF]; Palfrey and Rosenthal 1985; Myerson 2000; [START_REF] Dhillon | Economic theories of voter turnout[END_REF][START_REF] Feddersen | Rational choice theory and the paradox of not voting[END_REF]). Several alternative theoretical approaches [START_REF] Waas | Turnout[END_REF]) can be imagined, based on the empirical evidence that the decision to vote is correlated with an impressive set of variables [START_REF] Smets | The embarrassment of riches? A metaanalysis of individual-level research on voter turnout[END_REF]. In particular, both social [START_REF] Gerber | Social pressure and voter turnout: evidence from a large-scale field experiment[END_REF][START_REF] Rogers | Social pressure and voting: A field experiment conducted in a high-salience election[END_REF]) and ethical [START_REF] Blais | To Vote or Not to Vote? The Limits and Merits of Rational Choice Theory Pittsburgh[END_REF]; Blais and Achen 2019) motives appear important. The ethical motive for voting was first modelled as satisfaction derived from complying with a norm (Riker and Ordeshook 1968;[START_REF] Fiorina | The voting decision: instrumental and expressive aspects[END_REF][START_REF] Schuessler | Expressive voting[END_REF]. 1 Based on [START_REF] Harsanyi | Morality and the theory of rational behavior[END_REF][START_REF] Harsanyi | Rule utilitarianism, rights, obligations and the theory of rational behavior[END_REF], ethical voters were then modelled as civic-minded citizens who are rule utilitarians: they select a voting strategy that maximizes the total welfare of like-minded voters, which would obtain if all like-minded ethical voters selected it. Each ethical voter thus acts like a social planner for the group, generating positive equilibrium turnout rates that trade off the probability of winning against the group-aggregate cost of voting ( We propose a novel formalization of ethical voters, which imposes a less demanding ethical standard than rule utilitarianism by taking voters to have a mix of purely selfish and moral concerns. Moreover, the voters in our model are equipped with preferences -dubbed Homo moralis-shown to be plausible from an evolutionary perspective (Alger and Weibull 2013). 2 In a two-player game, a Homo moralis using strategy x when the opponent uses strategy y achieves utility

U (x, y) = (1 -κ) • π (x, y) + κ • π (x, x) ,
where π(x, y) is the individual's material payoff, given the strategies effectively used, and π(x, x) is the material payoff that the individual would obtain if, hypothetically, the opponent were instead to use the same strategy as him/her. The second term captures a Kantian moral concern, and κ ∈ [0, 1] is the individual's degree of morality. Immanuel [START_REF] Kant | Groundwork of the Metaphysics of Morals: A German-English Edition[END_REF] wrote "Act only according to that maxim whereby you can at the same time will that it should become a universal law." By maximizing the weighted sum of own material payoff and the Kantian moral concern, Homo moralis can be said to "act according to that maxim whereby you can at the same time will that others should do likewise with some probability." [START_REF] Alger | Homo moralis: preference evolution under incomplete information and assortative matching[END_REF] show that evolution favors Homo moralis preferences with κ equal to the probability with which individuals carrying rare mutant preferences get to interact. We take this result as our starting point and examine voters' decisions to participate in an election, assuming that they have Homo moralis preferences.

We model an election with two candidates. Prior to voting, each voter learns his own party preference but not the aggregate party preference distribution. Each voter chooses a costly level of effort, which determines the probability of managing to vote on time. In the ex ante setting (resp. ex post setting), voters select their voting strategies prior to (resp. after) learning their party preference. We model the electorate as a continuum, thus eliminating instrumental motives for voting. When contemplating a course of action, a Homo moralis evaluates what his material payoff would be if, hypothetically, a share κ of the population to which he belongs would follow the same course of action. 3 Our findings are as follows.

In the symmetric setting-where the candidates enjoy equally large expected supportfor any strictly positive κ there exists a unique equilibrium, with a strictly positive turnout rate (with one notable exception: the ex ante setting when the electorate is evenly divided in each state of Nature, not only in expected terms). With a convex effort cost function and a negligible marginal cost for small effort levels, voters can make a sizeable effort even if they attach a weight of only 0.1 or 0.2 to the Kantian moral concern. Somewhat surprisingly, however, a positive degree of morality is not sufficient to generate a positive turnout from the supporters of both parties: in asymmetric settings where one party is highly likely to be supported by a strong majority, voters who support the other party do not vote.

Whether in the symmetric ex ante or ex post setting, the equilibrium turnout rate is increasing in the degree of morality: by bolstering the hypothetical effect of his effort envisaged by a Homo moralis, an increase in κ raises the utility benefit of exerting costly effort.

Finally, the equilibrium turnout is lower in the ex ante than in the ex post setting. Behind the veil of ignorance as to his eventual party preference, a Homo moralis internalizes the effect that his voting effort has on both of his incarnations. This consideration is absent in the ex post scenario, where Homo moralis preferences make voters adopt a partisan standpoint.

The model

Consider an infinitely large population of voters, modelled as a continuum with mass 1. There are two political parties, A and B, and two states of Nature: in state ω A a majority q A ≥ 1/2 prefers A over B; in state ω B a majority q B ≥ 1/2 prefers B over A. State ω A occurs with probability a ∈ (0, 1) and state ω B with probability b = 1 -a. Prior to each election, Nature first draws the state and then a party preference P (i) ∈ {A, B} for each voter i. Each voter observes his realized preference but not the state of Nature.

Each voter cares about the margin with which his preferred party wins or loses. This assumption is natural for parliamentary elections, where margins determine the number of seats obtained. But it may also be reasonable in winner-take-all elections, for example because such margins have effects on the ability of candidates to raise funds. Formally, let v (δ) denote the material utility that a voter obtains from margin δ ∈ [-1, 1], where v is strictly increasing. Material utility encompasses both the utility associated with the concrete effects of the election outcome on laws, taxes, and government spending, and psychological benefits or costs derived from the relative power of the two parties. We take the cost of seeing one's preferred party lose by a margin δ to equal the benefit of seeing one's preferred party win by the same margin δ:

v (δ) = -v (-δ) for all δ ∈ [-1, 1] . (1)
Letting v be twice continuously differentiable and concave for δ > 0, v is then convex for δ < 0, and

v (δ) = v (-δ) and v (δ) = -v (-δ) . (2)
An example of such a function is

v (δ) = δ (2 + δ) (2 -δ) , (3)
for which v (δ) = 4 -3δ 2 and v (δ) = -6δ.

A voter who participates in the election votes for his preferred party. The probability that a voter reaches the voting booth on time and manages to vote is determined by his effort, a one-dimensional aggregate measure of time devoted to registering, efforts to overcome "obstacles" that may appear on the way to the voting booth (family and/or professional obligations, illness, inclement weather, etc), and attention devoted to avoiding behaviors that would annul the vote. Letting the probability of voting success be linear in effort, we let effort be a number between 0 and 1. The cost of exerting effort e ∈ [0, 1] is c (e), where c is a continuously differentiable and strictly convex function, with c (0) = 0 and c (1) = +∞.

Denote by e P (i) (i) ∈ [0, 1] the effort of a voter i with party preference P (i) ∈ {A, B}. Let the mass of voters be described by the interval [0, 1] with the Lebesgue measure. Because the allocation of preference to voters is random, the expected number of voters of type A who vote is:

1 0 1 P (i)=A e A (i) di.
Since, by assumption, the probability of being of a given type depends only on the state of nature, we will slightly abuse notation and write the above integral as follows in state ω A :

q A 0 e A (i) di.
With this notation in hand, party A's expected margin in state ω A is:

δ A = q A 0 e A (i) di - 1 q A e B (i) di, (4)
and party B's expected margin in state ω B is:

δ B = q B 0 e B (i) di - 1 q B e A (i) di. (5)
We will characterize type-homogenous equilibria, in which all the voters with the same party preference exert the same effort. If e A and e B denote these efforts, the expected margins in state A and B take the following simple forms:

δ A = q A e A -(1 -q A ) e B (6)
and

δ B = q B e B -(1 -q B ) e A . (7)
We distinguish between the ex ante and the ex post scenario, depending on whether each voter chooses a voting strategy before or after learning his party preference.

The ex ante scenario

In this scenario each voter i chooses a pair of efforts (e A (i) , e B (i)) ∈ [0, 1] 2 before learning his party preference P (i) ∈ {A, B}. Our goal is to characterize symmetric equilibria, where all voters choose the same effort pair. If all the other voters use some effort pair (e A , e B ), the expected material utility of a voter i who selects the effort pair (e A (i), e B (i)) is:

π (e A (i) , e B (i) , e A , e B ) = aq A v (δ A ) + b(1 -q B )v (-δ B ) -[aq A + b(1 -q B )] c (e A (i)) (8) +a(1 -q A )v (-δ A ) + bq B v (δ B ) -[a(1 -q A ) + bq B ] c (e B (i)) .
The first line is the expected material utility that the voter achieves if he turns out preferring party A, in which case he exerts effort e A (i) and enjoys material benefit v(δ A ) in state ω A and v(-δ B ) in state B. Since each voter has measure zero, the expected margins are indeed independent of i's efforts (e A (i), e B (i)). The second line is his expected material utility if he turns out preferring party B.

Prior to defining Homo moralis preferences, we state the following proposition (whose trivial proof is omitted).

Proposition 1 Voters with Homo oeconomicus preferences, represented by the expected material utility in [START_REF] Dhillon | Economic theories of voter turnout[END_REF], exert no voting effort.

Each voter being atomistic, his effort has no effect on the election outcome. Since voting is costly, purely instrumentally motivated voters do not participate. 4A voter with Homo Moralis preferences and degree of morality κ ∈ [0, 1] achieves the following utility from the effort pair (e A (i) , e B (i)) when the other voters use the effort pair (e A , e B ):

U (κ) (e A (i) , e B (i) , e A , e B ) = aq A v δ (κ) A + b(1 -q B )v -δ (κ) B -[aq A + b(1 -q B )] c (e A (i)) + a(1 -q A )v -δ (κ) A + bq B v δ (κ) B -[a(1 -q A ) + bq B ] c (e B (i)) , (9)
where δ (κ)

A = δ A + κ [q A (e A (i) -e A ) -(1 -q A )(e B (i) -e B )] (10)
and δ

(κ) B = δ B + κ [q B (e B (i) -e B ) -(1 -q B )(e A (i) -e A )] . (11)
The first line in [START_REF] Downs | An Economic Theory of Democracy[END_REF] is the voter's expected material payoff if he turns out preferring party A, evaluated at the expected margins δ (κ)

A and -δ

(κ)
B that would obtain if, hypothetically, a share κ of the other voters selected (e A (i) , e B (i)) instead of (e A , e B ). For example, if e A (i) > e A , the expected margin of party A would thus (hypothetically) increase by κq A [e A (i) -e A ] in state of nature ω A , while that of party B would be reduced by κ (1 -q B ) [e A (i) -e A ] in state of nature ω B . Because i's effort cost does not depend on the voting strategies of other voters, however, it is the true expected cost that appears in the utility in [START_REF] Downs | An Economic Theory of Democracy[END_REF]. The special case κ = 0 yields the familiar Homo oeconomicus preferences, examined above. The remainder of this section restricts attention to strictly positive degrees of morality κ ∈ (0, 1].

Using the symmetry of v (see ( 1)) we rewrite (9) as follows:

U (κ) (e A (i) , e B (i) , e A , e B ) = a(2q A -1)v δ (κ) A + b(2q B -1)v δ (κ) B (12) -[aq A + b(1 -q B )] c (e A (i)) -[a(1 -q A ) + bq B ] c (e B (i)) .
Hence, the marginal utilities of efforts e A (i) and e B (i) are, respectively:

∂U (κ) (e A (i) , e B (i) , e A , e B )
∂e

A (i) = κ aq A (2q A -1)v (δ (κ) A ) -b(1 -q B )(2q B -1)v (δ (κ) B ) -[ aq A + b(1 -q B ) ] c (e A (i)) (13) ∂U (κ) (e A (i) , e B (i) , e A , e B ) ∂e B (i) = κ bq B (2q B -1)v (δ (κ) B ) -a(1 -q A )(2q A -1)v (δ (κ) A ) -[ bq B + a(1 -q A ) ] c (e B (i)) . (14)
These equations immediately reveal that the expected marginal benefit (the first line in each equation) is nil if the voters are evenly split between the two parties in both states (q A = q B = 1/2); equilibrium efforts are then nil. The reason is clear: given the symmetry on v, if q A = q B = 1/2 the benefit a voter garners from making effort when incarnated as an A-supporter would be exactly offset by the reduction in benefit this effort generates for the voter incarnated as a B-supporter, and vice versa. Costly effort is therefore not warranted. Does this also mean that effort is nil in any setting where the voter is equally likely to end up as an A-supporter or a B-supporter (i.e., when a(2q

A -1) = (1 -a)(2q B - 1
))? No: the following proposition shows that, except in the special case where the voters are always evenly split (q A = q B = 1/2), a voter with Homo moralis preferences with positive degree of morality κ > 0 exerts a strictly positive effort, for at least one of the party preference realizations.

Proposition 2 For any degree of morality κ ∈ (0, 1], the zero-effort strategy profile (e * A , e * B ) = (0, 0) is a Nash equilibrium if and only if q A = q B = 1/2. This result obtains because the marginal cost is nil at zero effort and even the smallest strictly positive value of κ makes the voter evaluate the consequences of voting effort as if it had an impact on the outcome of the vote. Intuitive though as this result may seem, it is surprising that some strictly positive effort is undertaken even in the setting where the voter is equally likely to end up as an A-supporter or a B-supporter, i.e., when a(2q A -1) = (1 -a)(2q B -1). Indeed, as mentioned above, from an ex ante perspective any voting effort for one particular party then stands an equal chance of benefiting and harming the voter. Given that voting is costly, one could have expected that voters would then have preferred to refrain from voting. We will next deepen the analysis of the symmetric setting to better understand this result. Prior to this, however, we use a numerical example to show that a positive κ does not necessarily induce a positive effort for both party preference realizations. andc(e) = we/(1-e). With these functional forms, the algebraic computations can be handled by Mathematica. Take w = 2/10, q A = 8/10, and q B = 7/10. In the ex ante scenario:

Example 1 Let v (δ) = δ (2 + δ) (2 -δ)
1. For κ = 1/10 and a = 1/2, the only equilibrium is interior: (e *

A , e * B ) (.219, .142). We now analyze further the symmetric setting, in which both states of the world are equally likely and the share of voters who support party A in state ω A equals the share of voters who support party B in state ω B , i.e., a = 1/2 and q A = q B ≡ q. Given the previous proposition, we restrict attention to the non-trivial case where a strict majority of voters prefer party P in state ω P , i.e., q ∈ (1/2, 1]. In this setting each voter votes with a strictly positive probability, which does not depend on the voter's party preference.

If

Proposition 3 Consider a symmetric setting (a = 1/2, q A = q B ≡ q) in which q > 1/2.

For any degree of morality κ ∈ (0, 1], there exists a unique symmetric Nash equilibrium (e *

A , e * B ). This equilibrium entails a positive effort that is independent of party preference, i.e., e * A = e * B ≡ e * > 0.

To see why this result obtains, consider the equation that implicitly defines e * (see (37)):

κ(2q -1) 2 v (2q -1)e * = c (e * ). (15)
A positive voting effort e * makes the voters achieve a positive margin

δ * = (2q -1)e * (16)
for the majority at hand, a margin that generates an expected material benefit [q -(1q)]v(δ * ) to the voter (with probability q he is in the majority, and thus enjoys a positive margin of victory, but with the complementary probability he is in the minority and suffers from a loss). A positive degree of morality κ > 0 provides the voter with the satisfaction associated with the marginal effect κ(2q -1) that his level of effort would have on this margin, should a share κ of the other voters make the same effort. Hence, a costly effort is warranted.

The following proposition reports comparative statics results for the symmetric setting.

Proposition 4 Consider a symmetric setting (a = 1/2, q A = q B ≡ q). If v ≤ 0:

• for any q ∈ (1/2, 1], e * is strictly increasing in κ;

• for any κ ∈ (0, 1] there exists a threshold value q ∈ (1/2, 1] such that e * is strictly increasing in q for q ≤ q and strictly decreasing in q for q > q.

Ceteris paribus an increase in the degree of Kantian morality induces a higher participation rate, because an increase in κ bolsters the margin that would obtain if, hypothetically, a share κ of the other voters exerted the same effort as the voter at hand, and thus bolsters the satisfaction that this voter derives from exerting effort. By contrast, since the marginal cost of voting effort is increasing while its marginal material benefit is decreasing, an increase in the degree of partisanship q would not necessarily raise participation, for a given degree of morality κ.

The ex post scenario

In this scenario each voter i chooses an effort e P (i) (i) ∈ [0, 1] upon learning her party preference P (i) ∈ {A, B}. Our goal is to derive results for type-homogenous equilibria (ê A , êB ), where all the voters with a preference for party A (resp. B) exert the same effort êA (resp. êB ). By contrast to the ex ante analysis, here a voter updates her beliefs about the state of nature upon observing her party preference, and the material utility is evaluated given the realized party preference. Thus, given that all the other A-supporters exert effort e A and all the B-supporters exert effort e B , the expected material utility of an A-supporter i who exerts effort e A (i) is:

π (e A (i) , e A , e B ) = â • v (δ A ) + (1 -â) • v (-δ B ) -c (e A (i)) , (17)
where â is i's posterior belief that the state is ω A given that she prefers party A:

â = Pr [ω A |P (i) = A] = a • q A a • q A + b • (1 -q B ) . (18)
Likewise, the expected material utility of a B-supporter who exerts effort e B (i), given the efforts e A and e B exerted by the other voters, is:

π (e B (i) , e A , e B ) = b • v (δ B ) + 1 -b • v (-δ A ) -c (e B (i)) , ( 19
)
where b is her posterior belief that the state is favorable for party B, i.e., ω B :

b = Pr [ω B |P (i) = B] = b • q B b • q B + a • (1 -q A ) . ( 20
)
Before defining Homo moralis preferences, we observe that, as in the ex ante scenario, voters with Homo economicus preferences exert no voting effort (again the trivial proof is omitted).

Proposition 5 Voters with Homo oeconomicus preferences, represented by the expected material utilities in ( 17) and ( 19), exert no voting effort.

Turning now to Homo Moralis preferences, in the ex post scenario a voter with such preferences and a degree of morality κ ∈ (0, 1] evaluates what her expected material utility would be, should a share κ of the voters with the same party preference choose the same effort (instead of a share of all the voters as in the ex ante scenario). Formally, for any type-homogenous effort profile (e A , e B ) ∈ [0, 1] 2 used by the other voters, an A-supporter achieves the following utility from exerting effort e A (i):

U (κ) A (e A (i), e A , e B ) = â • v(δ (A,κ) A ) + (1 -â) • v(-δ (A,κ) B ) -c(e A (i)) ( 21
)
where δ

(A,κ) A = δ A + κq A [e A (i) -e A ], (22)
-δ

(A,κ) B = -δ B + κ(1 -q B )[e A (i) -e A ]. (23)
If a share κ of the other A-supporters were to choose the same effort as the individual 10 herself instead of their actual effort e A , the expected margin of party A would change by κq A [e A (i) -e A ] in state of nature ω A (see ( 22)), while the expected margin of party B would change by κ (1 -q B ) [e A (i) -e A ] in state of nature ω B (see [START_REF] Rogers | Social pressure and voting: A field experiment conducted in a high-salience election[END_REF]). As in the ex ante scenario, the utility cost of effort in ( 21) is unaffected by κ.

Likewise, a B-supporter achieves the following utility from exerting effort e B (i):

U (κ) B (e B (i), e A , e B ) = b • v(δ (B,κ) B ) + (1 -b) • v(-δ (B,κ) A ) -c(e B (i)) ( 24
)
where δ

(B,κ) B = δ B + κq B [e B (i) -e B ] (25)
and -δ

(B,κ) A = -δ A + κ(1 -q A )[e B (i) -e B ]. (26)
A type-homogenous equilibrium (ê A , êB ) solves the following fixed-point problem:

   êA ∈ arg max e A (i)∈[0,1] U (κ) A (e A (i), êA , êB ) êB ∈ arg max e B (i)∈[0,1] U (κ) B (e B (i), êA , êB ). ( 27
)
We first prove that in any such equilibrium (if it exists), all voters exert a positive effort.

To see this, consider first the marginal utility of effort of a voter i with party preference

P (i) = A: κ q A âv δ (A,κ) A + (1 -q B ) (1 -â) v -δ (A,κ) B -c (e A (i)) . ( 28
)
For any type-homogenous efforts e A and e B exerted by the other voters, the assumptions v (•) > 0 and c (0) = 0 imply that when evaluated at e A (i) = 0, this expression is strictly positive for any κ ∈ (0, 1]. Since the same remark applies to the marginal utility of a voter i with party preference P (i) = B, we have proved the following result.

Proposition 6 For any degree of morality κ ∈ (0, 1], any type-homogenous equilibrium (ê A , êB ) is such that all the voters exert a positive effort: êA > 0 and êB > 0.

This result contrasts with the ex ante scenario, in which a positive degree of morality is not sufficient to generate a strictly positive effort for all parameter constellations.

Example 2 For the same parameters as in Example 1,in the ex post scenario:

1. For κ = 1/10 and a = 1/2, (ê A , êB ) (.339, .320).

2. For κ = 1 and a = 1/2, (ê A , êB ) (.718, .711).

3. For κ = 1/10 and a = 3/4, (ê A , êB ) (.359, .280).

Equilibrium existence is more challenging to prove than in the ex ante scenario, and we had to add some conditions to do so; these additional conditions are, however, not necessary to prove that if an equilibrium exists it is unique, as shown in the following proposition.

Proposition 7 Consider a symmetric setting (a = 1/2, q A = q B ≡ q ≥ 1/2). For any κ ∈ (0, 1] there exists a unique candidate for a symmetric type-homogenous equilibrium (ê A , êB ) = (ê, ê). For (ê, ê) to be an equilibrium it is sufficient that v (δ) ≤ 0 for δ ∈ [0, 1] and c (e) ≥ 0 for all e ∈ [0, 1].

Prior to comparing the equilibrium effort in the ex post scenario to that in the ex ante scenario, we show that the comparative statics are qualitatively the same as in the ex ante scenario.

Proposition 8 Consider a symmetric setting (a = 1/2, q A = q B ≡ q). If v ≤ 0:

• for any q ∈ (1/2, 1], ê is strictly increasing in κ;

• for any κ ∈ (0, 1] there exists a threshold value q ∈ (1/2, 1] such that ê is strictly increasing in q for q ≤ q and strictly decreasing in q for q > q.

Propositions 6 and 7 imply that if an equilibrium exists in the symmetric case, it is implicitly defined by this equation:

κ q 2 + (1 -q) 2 v (2q -1)ê = c (ê) . (29)
As in the ex ante scenario, a positive voting effort ê makes the independent voters achieve a positive margin δ = (2q -1)ê (30)

for the majority at hand. However, the marginal benefit of this margin is not the same as in the ex ante setting. For the sake of illustration consider a voter who prefers party A.

The term [q 2 + (1 -q) 2 ] in (29) captures both the benefit of increasing the margin of victory when A is the majority party and the benefit of decreasing the victory margin when B is the majority party. Indeed, in the symmetric setting, the posterior belief of this voter that the state is ω A is â = q. Hence, κq 2 v δ is the marginal expected utility benefit from increasing the victory margin in state ω A (in which an effort increase raises the hypothetical margin that Homo moralis ponders by κq). Similarly, κ(1 -q) 2 v δ is the marginal expected utility benefit for the A-supporter at hand from decreasing the victory margin of party B in state ω B (in which an effort increase reduces the hypothetical margin envisaged by Homo moralis by κ(1 -q)).

Intuition suggests that voters exert more effort in the ex post than in the ex ante scenario, since in the latter the voters internalize the fact that they may end up preferring either party, whereas in the former preferences are purely partisan. A glance at the equations that implicitly define e * (see [START_REF] Harsanyi | Morality and the theory of rational behavior[END_REF]) and ê (see (29)) reveals that the comparison hinges on the terms q 2 + (1 -q) 2 and (2q -1) 2 . Since q 2 + (1 -q) 2 > (2q -1) 2 for any q ∈ [1/2, 1], we immediately obtain the following result, which confirms the intuition.

Proposition 9 Consider a symmetric setting (a = 1/2, q A = q B ≥ 1/2). For any degree of morality κ ∈ (0, 1], the equilibrium effort ê in the ex post scenario strictly exceeds the equilibrium effort e * in the ex ante scenario.

Conclusion

Our exploratory analysis shows that the evolutionarily plausible class of Homo moralis preferences can lead to high turnout rates in large elections. Moreover, equilibrium turnout rates depend on the specifics of the election in hand. Since these preferences may be interpreted as capturing a partially Kantian motivation, the model is arguably plausible also in light of the ethical motive for voting often put forward by voters in surveys [START_REF] Blais | To Vote or Not to Vote? The Limits and Merits of Rational Choice Theory Pittsburgh[END_REF].

Appendix

Proof of Proposition 2

Consider a voter who expects the other voters to exert no effort, (e A , e B ) = 0. This implies that when evaluated at e A (i) = e B (i) = 0, δ

(κ) A = 0 = δ (κ) B , so that v (δ (κ) A ) = v (δ (κ) B ).
Simplification of ( 13) and ( 14) then lead to the following necessary and sufficient conditions for (e *

A , e * B ) = (0, 0) to be a Nash equilibrium:

aq A (2q A -1) -b(1 -q B )(2q B -1) ≤ 0 (31)
and bq B (2q B -1) -a(1 -q A )(2q A -1) ≤ 0.

(32)

Proof of Proposition 4

By some abuse of notation, let e * denote the function that to each pair (κ, q) ∈ (0, 1]×(1/2, 1] associates the value e * implicitly defined by [START_REF] Harsanyi | Morality and the theory of rational behavior[END_REF]. By the implicit function theorem, in a neighborhood of e * :

∂e * (κ, q) ∂κ = (2q -1) 2 v (δ * ) c (e * ) -κ 2 (2q -1) 3 v (δ * ) , (38)
which is strictly positive,

∂e * (κ, q) ∂q = 4κ(2q -1)v (δ * ) + 2κ 2 (2q -1) 2 e * v (δ * ) c (e * ) -κ 2 (2q -1) 3 v (δ * ) (39) = 2κ(2q -1) c (e * ) -κ 2 (2q -1) 3 v (δ * ) • [2v (δ * ) + (2q -1)e * v (δ * )] ,
and

∂e * (κ, q) ∂ = κ(2q -1) 2 v (δ * ) + κ 2 (2q -1) 3 e * v (δ * ) c (e * ) -κ 2 (2q -1) 3 v (δ * ) (40) = κ(2q -1) 2 c (e * ) -κ 2 (2q -1) 3 v (δ * ) • v (δ * ) + 2 (2q -1)e * v (δ * ) .
The denominator being strictly positive,

sign ∂e * (κ, q) ∂q = sign [2v (δ * ) + (2q -1)e * v (δ * )] . (41)
Since (41) is strictly positive for q = 1/2, by continuity it is also strictly positive for q close to 1/2. For e * to be increasing in q, the expression in (41) must be positive for q ∈ [1/2, 1]. Since δ * is increasing in e * and v is strictly concave, the first term would then decrease in q, while the absolute value of the second term would increase in q. Hence, there may exist some q ∈ (1/2, 1] such that the sum of the two terms is nil for q = q. We show, by contradiction, that e * must then be decreasing in q for q > q. Thus, consider some q > q and let ê * denote the associated equilibrium effort. Suppose that there exists some ε > 0 such that e * ≥ ê * for all q ∈ [q, q + ε]. But then (41) would be strictly negative for q ∈ (q, q + ε], contradicting the assumption that e * ≥ ê * for all q ∈ [q, q + ε]. Q.E.D.

Proof of Proposition 7

In the symmetric setting, â = b = q. Consider a voter i with party preference P (i), and assume that all the other voters make effort e * ∈ (0, 1). Then δ A = δ B = (2q -1) e * ≡ δ, and i's expected utility as a function of e P (i) (i) is:

U (κ) P (i) e P (i) (i) , e * , e * = q • v δ + κq e P (i) (i) -e * (42) + (1 -q) • v -δ + κ (1 -q) e P (i) (i) -e * -c e P (i) (i) .
e * is a symmetric equilibrium iff e P (i) (i) = e * maximizes the utility in (42). Since ∂U (κ)

P (i) e P (i) (i) , e * , e * ∂e P (i) (i) = κq 2 • v δ + κq e P (i) (i) -e * (43) +κ (1 -q) 2 • v -δ + κ (1 -q) e P (i) (i) -e * -c e P (i) (i) ,
replacing e P (i) (i) by e * in (43) and equating the resulting expression to zero gives the necessary first-order condition for e * to be a symmetric interior equilibrium. By symmetry of v (see ( 2)), this condition reduces to:

κ[q 2 + (1 -q) 2 ] • v ((2q -1) e * ) = c (e * ) . (44)
The right-hand side is a strictly increasing function that equals 0 for e * = 0 and tends to +∞ as e * tends to 1. For any κ > 0 the left-hand side is strictly positive for e * = 0, and it is strictly decreasing since its derivative wrt e * ,

κ 2 (2q -1) [q 2 + (1 -q) 2 ] • v ((2q -1) e * ) ,
is strictly negative. We conclude that (44) has a unique interior solution e * ∈ (0, 1). To prove the second statement in the proposition, we turn to the second partial derivative:

∂ 2 U (κ) P (i) e P (i) (i) , e * , e * ∂ e P (i) (i) 2 = (κ) 2 q 3 • v δ + κq e P (i) (i) -e * ( 45
) (κ) 2 (1 -q) 3 • v -δ + κ (1 -q) e P (i) (i) -e * -c e P (i) (i) .
When evaluated at e P (i

) (i) = e * this equals (κ) 2 [q 3 -(1 -q) 3 ] • v ((2q -1) e * ) -c (e * ) .
This is strictly negative; to see this, note that q 3 > (1 -q) 3 and that v is strictly negative since the argument (2q -1) e * is strictly positive, and recall that c > 0). Hence, we can first conclude that e P (i) (i) = e * is a local maximum. However, for e * to be an equilibrium it must also be a global maximum. This is what we show next.

e * is a global maximum if the second derivative in (45) is negative for all e P (i) (i) ∈ [0, 1]. Since c > 0 and (κ) 2 > 0, a sufficient condition for this to be true is: q 3 • v δ * + κq e P (i) (i) -e * + (1 -q) 3 • v -δ * + κ (1 -q) e P (i) (i) -e * < 0, or q 3 • v (x) + (1 -q) 3 • v -x -κ e * -e P (i) (i) < 0 (46) where x = (2q -1)e * + κq e P (i) (i) -e * .

(47)

A priori, condition (46) may fail to hold. To see this, let κ = 1 and e P (i) (i) = 0; then x < 0, and it follows that v (x) > 0 and v -x -κ e * -e P (i) (i) = v (-qe * ) > 0. Although the second partial derivative in (45) may thus be positive for some e P (i) (i) ∈ [0, 1], we will now show that it changes sign at most once. This implies that for any given e * , U is quasi-concave in e P (i) (i) for e P (i) (i) ∈ [0, 1], a property which ensures that the local maximum e P (i) (i) = e * is also a global maximum in [0, 1].

For U to be quasi-concave, it suffices that the second partial derivative in (45) be decreasing in e P (i) (i). If c (e) ≥ 0 for all e ∈ [0, 1] (one of the conditions stated in the proposition), it suffices that the left-hand side of (46) is decreasing in e P (i) (i) for this to be true. If v (e) ≥ 0 (the other condition stated in the proposition), then the left-hand side of (46) is decreasing in e P (i) (i) if the arguments of v (that is, x and -x-κ[e * -e P (i) (i)]) are increasing in e P (i) (i). This is clearly true, as revealed by inspection of the expression for x in (47) and the fact that -x -κ[e * -e P (i) (i)] = -(2q -1)e * + κ(1 -q)[e P (i) (i) -e * ]. Q.E.D.

Proof of Proposition 8

By some abuse of notation, let ê denote the function that to each pair (κ, q) ∈ (0, 1] × [1/2, 1] associates the value ê implicitly defined by (29). By the implicit function theorem, in a neighborhood of ê:

∂ê(κ, q) ∂κ = [q 2 + (1 -q) 2 ] v ( δ) c (ê) -κ 2 (2q -1) [q 2 + (1 -q) 2 ] v ( δ) , (48)
which is strictly positive, and ∂ê(κ, q) ∂q = κ [2q -2(1 -q)] v ( δ) + 2κ 2 (2q -1) [q 2 + (1 -q) 2 ] êv ( δ) c (ê) -κ(2q -1) [q 2 + (1 -q) 2 ] v ( δ) (49) = 2κ(2q -1) c (ê) -κ 2 (2q -1) 3 v ( δ)

• v ( δ) + q 2 + (1 -q) 2 êv ( δ) .

The denominator being strictly positive, for any q > 1/2, we have sign ∂ê(κ, q) ∂q = sign v ( δ) + q 2 + (1 -q) 2 êv ( δ) .

(50)

Application of the same logic as the one used in the proof of Proposition 4 leads to the stated result. Q.E.D.

  Coate and Conlin 2002; Feddersen and Sandroni 2006; Feddersen et al 2009; Krishna and Morgan 2015; Bierbrauer et al. 2019).

  κ increases, turnout increases and for κ = 1 and a = 1/2 it reaches (e * A , e * B ) (.613, .535).

3 .

 3 If the asymmetry increases sufficiently, turnout for one of the parties vanishes. For κ = 1/10 and a = 3/4, (e * A , e * B ) (.282, 0).

A voter's satisfaction may also stem from helping sustain democracy or expressing political views.

Our approach is thus close in spirit to[START_REF] Conley | Memetics and voting: how nature may make us public spirited[END_REF]

We refer to our companion paper (Alger and Laslier, 2020) for an explanation for why the utility of Homo moralis in a continuum population takes this form.

For the same reason, any consequentialistic utility function-even if it represents altruistic inclination towards the other voters rather than pure material self-interest-would generate zero effort.

* I.A. acknowledges funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 789111 -ERC EvolvingEconomics), as well as IAST funding from the French National Research Agency (ANR) under grant ANR-17-EURE-0010 (Investissements d'Avenir program). We are grateful for comments from audiences at at the Paris School of Economics, the Social Choice and Welfare conference in Seoul, the COMSOC online seminar, and the UC Louvain Hoover Chair. Olivier Lision provided excellent research assistance.

The statement in the proposition then follows from three observations. First, q A q B > (1q A )(1 -q B ) implies that at least one of the conditions is violated (to see this, note that aq

Second, since q A ≥ 1/2 and q B ≥ 1/2, q A q B ≤ (1 -q A )(1 -q B ) if and only if q A = q B = 1/2. Third, the two necessary and sufficient conditions are satisfied for q A = q B = 1/2. Q.E.D.

Proof of Proposition 3

Since q > 1/2, Proposition 2 implies that either e * A > 0 or e * B > 0. Assume that there exists an equilibrium such that e *

A > e * B ≥ 0. At such an equilibrium, the necessary first-order condition for e * A is obtained by equating the expression in ( 13), evaluated at e A (i) = e * A and e B (i) = e * B , to zero. Since the term multiplying κ in δ

A and in δ (κ)

B (see [START_REF] Feddersen | Rational choice theory and the paradox of not voting[END_REF] and ( 11)) then equals zero, the first-order condition reduces to

where

Likewise, e * B ∈ [0, e * A ) must satisfy the inequality

Now,

. Hence, the left-hand side of (36) strictly exceeds that of (33). But together with strict convexity of c, this contradicts e * A > e * B . Having thus proved that at any equilibrium e * A = e * B ≡ e * > 0, any such e * satisfies the necessary first-order condition κ(2q -1) 2 v ((2q -1)e * ) = c (e * ).

(37)

Existence and uniqueness follow from our assumptions on v and c. Q.E.D.


D Gardiol

D Barghini

A Buzzoni

A Carbognani

M Di Carlo

M Di

C Knapic

E Londero

G Pratesi

S Rasetti

W Riva

R Salerno

G M Stirpe

G B Valsecchi

C A Volpicelli

S Zorba

F Colas

B Zanda

S Bouley

S Jeanne

A Malgoyre

M Birlan

C Blanpain

J Gattacceca

J Lecubin

C Marmo

J L Rault

J Vaubaillon

P Vernazza

F Affaticati

M Albani

A Andreis

G Ascione

T Avoscan

P Bacci

R Baldini

A Balestrero

S Basso

R Bellitto

M Belluso

C Benna

F Bernardi

M E Bertaina

L Betti

R Bonino

K Boros

A Bussi

C Carli

T Carriero

E Cascone

C Cattaneo

A Cellino

P Colombetti

E Colombi

M Costa

G Cremonese

D Cricchio

G D'agostino

M. D 'elia

M De Maio

P Demaria

A Di Dato

R Di Luca

F Federici

V Gagliarducci

A Gerardi

G Giuli

D Guidetti

G Interrante

M Lazzarin

S Lera

G Leto

D Licchelli

F Lippolis

F Manca

S Mancuso

F Mannucci

R Masi

S Masiero

S Meucci

A Misiano

V Moggi Cecchi

E Molinari

J Monari

M Montemaggi

M Montesarchio

G Monti

P Morini

A Nastasi

E Pace

R Pardini

M Pavone

A Pegoraro

S Pietronave

T Pisanu

N Pugno

U Repetti

M Rigoni

N Rizzi

C Romeni

M Romeo

S Rubinetti

P Russo

F Salvati

D Selvestrel

R Serra

C Simoncelli

R Smareglia

M Soldi

R Stanga

F Strafella

M Suvieri

C Taricco

G Tigani Sava

M Tombelli

P Trivero

G Umbriaco

R Vairetti

G Valente

P Volpini

R Zagarella

A Zollo

Cavezzo meteorite recovery 1 Cavezzo, the first Italian meteorite recovered by the PRISMA fireball network. Orbit, trajectory, and strewn-field

Keywords: meteorites, meteors, meteoroids -methods: data analysis -techniques: image processing

INTRODUCTION

The analysis of meteoritic material plays a relevant role in modern planetary sciences, since the fall of meteorites provides the easiest and cheapest way to gather extra-terrestrial samples. The mineralogy and petrology of these samples are the major sources of information about the geology, formation and evolution of minor and major bodies in the Solar System and beyond (e.g. Kruijer et al. 2020 and references therein). The scientific importance of such material is even higher if the interaction of the body with the Earth's atmosphere, generating the meteorite falls, is observed. Firstly, the observation of the meteor can provide crucial information about the physical properties of the body entering the atmosphere, which usually ablates for the most part before reaching the ground, if anything survives. In this case, the knowledge of the orbit of the meteoroid unveils its origin in the Solar System which can be linked with the physical and chemical characterisation of the meteorite itself. At this time of writing, the Meteoritical Database 1 lists almost 64,000 officially classified meteorites. Among them, to the authors' knowledge, only 35 (plus at least 2 not yet published) were collected together with a sufficient set of observations of their atmospheric path, allowing a reliable reconstruction of their heliocentric orbit before the interaction with the Earth's atmosphere. Gathering sufficient statistics for meteoroid orbits would enable investigations into the possible link between different meteorite classes and their origin in the Solar System. Ultimately, the knowledge of the source regions of particular meteorite groups can provide constraints for the identification of a common parent body. As an example of recent relevant results on this topic, the reader can refer to Granvik & Brown (2018), Jenniskens et al. (2019) and Unsalan et al. (2019). The importance of such results for planetary science is so pronounced that efforts even have been made to reconstruct meteorite pre-impact orbits from historical records (Gounelle et al. 2006).

The case of the Pribram meteorite in 1959 (Ceplecha 1961) represents the first successful meteorite recovery resulting from the observation of a bright meteor, which allowed for precise computation of its atmospheric trajectory, dynamics and dark flight. At the same time, this was the first meteorite recovery carried out thanks to a systematic meteors observation survey. Following the example of the Czechoslovakian Fireball Network, now European Fireball Network (Spurný et al. 2017b), many dedicated projects started to realise observational networks. Run by both amateur and professional astronomers, these networks have a shared goal of continuously monitoring the night-sky and detecting meteor and fireball events. The scientific outcome for this kind of survey is twofold; providing a unique tool to discover new meteor showers by focusing on the faint but predominant component of the detected events, and capturing the very rare occurrence of meteorite-dropping fireballs. In this context, one should highlight that only 20 among the 35 "pedigree" meteorites were collected as a result of dedicated 1 The Meteoritical Society, https://www.lpi.usra.edu/meteor/ observational surveys (see Tab. 1). Excluding the well-known but out-of-range events of Almahata Sitta and Chelyabinsk, the remaining 13 falls were documented through sporadic observations only, such as security cameras, dash cams and visual reports. For this reason, the past few years have witnessed a remarkable and ever increasing effort to extend the coverage of meteor networks worldwide and maximise the efficiency in the recovery of meteorites. As a result, 9 among the 20 meteorites collected, thanks to meteor surveillance networks, were recovered most recently between 2014 -2020.

In this international scenario, the PRISMA all-sky camera network [START_REF] Gardiol | Proceedings of the 35th International Meteor Conference Egmond, the Netherlands[END_REF][START_REF] Gardiol | Proceedings of the 37th International Meteor Conference[END_REF] was born in 2016 to achieve a systematic surveillance of meteors and fireballs in the skies over the Italian territory. In fact, PRISMA stands for "Prima Rete Italiana per la Sorveglianza sistematica di Meteore e Atmosfera" (First Italian Network for Meteors and Atmosphere systematic Surveillance). At this time of writing, PRISMA has deployed 52 stations, among which 37 are fully operating and 15 are in installation phase. The PRISMA project is part of the international collaboration initiated by the FRIPON project (Fireball Recovery and InterPlanetary Observation Network, [START_REF] Colas | Proceedings of the 33th International Meteor Conference[END_REF], Jeanne et al. 2019and Colas et al. 2020, submitted).

In this paper, we report on the finding of two meteorite pieces in Italy, near the Cavezzo village (Modena, Emilia-Romagna). The meteorite-dropping fireball, which reached a brightest absolute magnitude of -9.5, was observed on the evening of New Year's Day 2020 by eight all-sky cameras of the PRISMA network. The two fragments, weighing 3.1 g and 52.2 g respectively, were collected as a result of a dedicated field search and thanks to the involvement of the local people. In Sect. 2, we illustrate the preliminary strewn-field computation and the meteorite search activity, and give a short description of the two recovered fragments. Section 3 gives a complete review of the fireball data analysis and its physical characterisation. In Sect. 4, we provide the orbital parameters and discuss a possible progenitor for the observed meteoroid. We draw our conclusions in Sect. 5.

THE FIREBALL EVENT

Preliminary trajectory and strewn-field

On 1 st January at 18:26:53 UT eight stations of the PRISMA network detected a brilliant fireball, named IT20200101, in the skies of northern Italy (the list is shown in Tab. 2). The FRIPON automatic alert system performed a preliminary data reduction based on four cameras (ITPI03, ITTO02, ITER04, ITVE02). The remaining cameras were not configured to work as part of the automatic data reduction pipeline, having had being installed just prior to the detection. These preliminary results indicated a high probability of a meteorite-dropping fireball, as the computed preatmospheric velocity was found to be about 12 km s -1 and the inclination of the trajectory was high (68 • ). The meteor was tracked down to a height of about 22 km, and the light curve showed two sudden brightenings at altitude of about 32 km and 30 km. In order to get a preliminary estimate of the strewn-field, we reprocessed the data manually, adding two of the missing cameras (ITLO03 and ITER01). In the meantime, we also started to receive many reports

O R I G I N A L U N E D I T E D M A N U S C R I P T
Cavezzo meteorite recovery 3 The high angle of fall resulted in an intense ablation process that led the fireball to shine with a mean absolute visual magnitude of about -7.5. The fireball disappeared from the camera images at an altitude of about 21.5 km. According to the first computation of the strewn-field, made using a purely ablative model, we estimated a nominal impact point close to the village of Disvetro in the municipality of Cavezzo (province of Modena), near the local astronomical observatory, in the middle of the Po Valley. However, the light curve profile suggested that the object underwent a fragmentation process during the atmospheric flight, as also supported by eyewitnesses. Therefore, we expected meteorite pieces to be spread around the line joining Disvetro with the on-ground projection of the final part of the visible trajectory, near the village of Rovereto sulla Secchia.

Meteorite search and recovery

As soon as the preliminary strewn-field was identified (the day following the fall, i.e., 2 nd January in the early afternoon) we had to decide a strategy for the meteorite search. Within the PRISMA collaboration, a team of volunteers, both professionals and amateurs, is trained for meteorite hunting and dedicated to search activities. The strewn-field, located between the towns of Rovereto sulla Secchia and Disvetro, lies in a rural territory with many cultivated fields, groves and houses spread over the area. Since the vast majority of the terrain is private property, we notified the local authorities that teams of hunters would be there to search for meteorites on behalf of the PRISMA collaboration. To help searchers enter private terrains and areas, we also involved the Civil Protection of Cavezzo, as they are usually employed during public events, and their members are well known among the community. At the same time we prepared a press release to be published on the PRISMA website3 and on the outreach platform of INAF (MediaINAF4 ). In the press release, we provided all the necessary information about the most probable area where fragments could be found and also a brief tutorial on how to recognise a freshly fallen meteorite. The goal of this strategy was to maximise the probability of a successful recovery by involving a larger number of people, even if not specially trained. The press release was also sent to local media in the Modena area, encouraging local inhabitants to start their own search or at least to be aware that they might chance upon meteorite fragments, and in this case to contact us by email. The news was made public on the late afternoon of 2 nd January, and already on 3 rd January we started receiving reports from people in the local area and giving interviews to local and national newspapers and televisions. By the morning of 4 th January, a team of about twenty hunters was ready to start searches on Sunday 5 th , while a small scouting group from Bologna University led by Romano Serra was already onsite. On 4 th January at 3 PM local time we received an email with the first reliable meteorite candidate from Mr. Davide Gaddi, reporting the recovery of a small fragment (the size of a fingernail) on an embankment along the Secchia river. We immediately arranged a meeting with Romano Serra, where Mr. Gaddi showed also a second larger fragment, found in the meantime in the same place, the size of a walnut. Both fragments were recognised to be freshly fallen meteorites. It was less than three days since the fireball event.

In the following days, the search for other fragments carried out by teams organised by PRISMA and by other people did not lead to further findings, even though a refined strewn-field was available in the meantime 5 . Bad weather and muddy fields also limited the area accessible for the search. We planned to resume searches in Spring 2020 hoping for better weather and terrain conditions but, up until now, the COVID-19 pandemic outbreak did not allow us to organise further on-field activities.

Meteorite fragments description

The two fragments were recovered at coordinates 44 • 49 ′ 43. ′′ 7 N 10 • 58 ′ 19. ′′ 5 E, at the border of a narrow country road that runs parallel to the left main embankment of the Secchia river. This is approximately halfway between Villa Motta and Rovereto sulla Secchia in the territory of the municipality of Cavezzo. Figure 1 shows the two finds. Fragment n.1 (F1), the first one to be found, has a tetrahedral form and weighs 3.1 g. Fragment n.2 (F2), the largest one, weighs 52.2 g. Both fragments clearly present a recently formed fusion crust on most of their surface. They also show what appears to be a light grey chondritic pattern on one of their sides, which was probably due to fragmentation that most likely occurred when they hit the ground. This interpretation is corroborated by the presence of an impact feature that is visible on one of the edges separating two of their sides, also accompanied by white streaks (Fig. 2a).

In addition, F2 presents a darker grey colouration on one side, apparently a less pronounced secondary fusion crust, which is compatible with exposure due to a fragmentation event most probably associated to the brightening which occurred at a height of around 30 km. The meteorite fragments are currently being analysed at the Department for Earth Science of the Firenze Uni-5 PRISMA and FRIPON teams independently computed two strewn-fields that are in close agreement with each other.

versity for classification and study of the mineralogical, petrographic and geochemical characteristics. The result of this extensive analysis will be the subject of a dedicated article (Pratesi et al. 2020, submitted). However, as for fragment F2, it is worth mentioning that the composition of olivine and low-Ca pyroxene, the abundances of the mineral phases and the texture (Fig. 2b) are fully compatible with the L chondritic group. On the other hand, the modal mineralogy of fragment F1 (namely, the extremely low content of iron and troilite and the very high abundance of high-Ca pyroxene) is clearly different from the one typically found in L chondrites. On 5 th September 2020, the Nomenclature Committee of the Meteoritical Society has approved the classification proposal of "L5 anomalous" chondrite. Cavezzo is the only approved meteorite belonging to this class 6 . Figure 2b shows a photomosaic of polarising optical microscope images of a thin section obtained from F2, where, distributed in the matrix, chondrules and chondrule fragments can be observed.

The γ-ray activity measurement performed on the F2 sample at the INAF Monte dei Cappuccini Laboratory in Torino (Taricco et al. 2006, Colombetti et al. 2013) has shown the presence of many cosmogenic radioisotopes. Despite the small mass of F2 with respect to the samples that are commonly measured at this facility, the two characteristic lines of 48 V at 983.53 and 1312.11 keV7 are clearly visible, confirming beyond any doubt the presence of this radioisotope. Since 48 V has a half-life of 15.97 d, this is an indisputable proof of the very recent fall of the recovered meteorite, thus linking it again to the New Year's fireball. The results of the radiometric measures of F2 will be the subject of a forthcoming publication.

FIREBALL DATA ANALYSIS

PRISMA, as a partner of the FRIPON collaboration, currently shares the same technology of the network. Each station is equipped with a CCD camera (6 mm diagonal, 1296 x 966 px), coupled with a short focal lens objective (1.25 mm), to obtain an all-sky field of view (FOV). The camera is connected to a Linux operating mini-PC via LAN and controlled by the open-source FREETURE software [START_REF] Audureau | Proceedings of the 33th International Meteor Conference[END_REF]). The camera is operated at 30 fps (1/30 s exposure time) in order to sample the meteor trail with a suitable rate. The meteor detection is triggered locally in each node by a frame difference method, and cross-correlated with respect to data of other nodes to check for multiple detections of the same event. The 30 fps video stream of the detected meteor is saved locally as FITS files and, in the case of a multiple detection, it is collected by the FRIPON central server, located at the Laboratoire The first step in the analysis of meteor detections is the astrometric and photometric reduction. Since almost no stars arise from the background noise in the 30 fps video stream, the control software acquires a 5 s exposure image, named capture, every 10 minutes. With a limiting V magnitude of about +4.5 on this set of images, available for each operational night, hundreds of stars per frame are automatically identified and correlated with catalogue positions. They are then used as a reference for both astrometric and photometric calibration. Concerning astrometry, we implemented the approach introduced by Ceplecha (1987), [START_REF] Borovička | [END_REF] and Borovička et al. (1995), in which the absolute astrometric solution of the camera is derived, in the alt/az system, as a function of 8 parameters. This analytical description accounts for the major distortion factors that are, in the case of PRISMA all-sky cameras, the pronounced radial distortion of the fish-eye lens and the possible mismatch of the optical centre with respect to the local zenith direction. Two additional parameters are introduced if the optical plate is found to be misaligned with respect to the local horizon plane. With respect to Borovička et al. (1995), we provided a new explicit parametrisation of the astrometric model, which reduces the parameters' correlation degree and improves overall convergence properties for the determination of the astrometric solution (Barghini et al. 2019a). At the same time, the photometric calibration is determined on the same set of images as well. Since no filter is applied over the lens, a wide passband magnitude is considered, roughly between 400 and 800 nm, on the basis of the quantum efficiency of the camera and the transmission of the glass dome. This panchromatic magnitude is numerically computed from the catalogue UBVRI Johnson-Cousins system and therefore used to derive the zero-point and atmospheric extinction coefficient for each capture. For PRISMA cameras, we also determined and accounted for the efficiency loss along the radial direction that turns out to be of about 40 per cent from the centre to the very edge of the camera (Barghini et al. 2019b). We used the method outlined in Carbognani et al. (2020) to estimate the fireball atmospheric trajectory, its main physical parameters, and the best kinematic parameters in the terminal point of the luminous path. The atmospheric trajectory computation is performed according to the classical formulation reported in Ceplecha (1987) and [START_REF] Borovička | [END_REF]. The parameters of the dynamical model are derived based on the description given by Kalenichenko (2006) and references therein. In our case, however, the meteoroid has been most probably subject to fragmentation phenomena, so that the physical quantities are to be taken with caution. For the dark flight and strewn-field determination, Ceplecha (1987) formulation is again used, by computing the expected impact points in a range of mass-section ratios. The orbital parameters are derived both in an analytical way and by numerical integration, as the two methods have proven to provide consistent results [START_REF] Clark | [END_REF].

Astrometry and photometry

Figure 3 shows the map of PRISMA stations that detected the IT20200101 fireball (white dots) together with the onground projection of the reconstructed trajectory (red line) and the images of the meteor trail seen by each camera (enclosed in white circles) obtained through the analysis of video records. The distance between the eight stations and the fireball atmospheric trajectory spans between 75 and 200 km. While some cameras captured the bright flight in the central area of the FOV, other cameras recorded the fireball at a quite low elevation above the horizon. In particular, the astrometric reduction for data of two cameras (Felizzano and Cecima) results in elevations lower than 10 • for the final 5-10 points of the trajectory. Unfortunately, sky quality condition is not optimal in many PRISMA observational sites, due to light pollution which is especially severe in the Po valley. These experimental constraints prevent us from detecting stars, below 10 • of elevation for most PRISMA cameras, even in long-exposure calibration images. While residual systematic effects are numerically addressed (Barghini et al. 2019b), this correction can be only tentative below 10 • of elevation and the positional accuracy for these last points is questionable. To assess the potential effect of this bias over the final result, we first excluded these points and verified that the overall results (i.e., the trajectory computation) were unchanged within the measurement errors. In fact, their total weights are not predominant over other reference points, at the same timing, from the remaining six cameras. We finally included them since they provide important photometric data for the trailing edge of the fireball light curve.

The photometric analysis highlights that the fireball point spread function (PSF) saturated, in almost all cameras, between 2 s and 4.7 s from the beginning of the bright flight. Barghini et al. (2019b) give a comprehensive analysis of the effects of PSF saturation for PRISMA cameras concluding that the astrometric precision is not significantly degraded at least below h ≤ 4, where h is the relative ratio of the PSF height to the saturation value, namely 4095 ADU (analog-to-digital units) for PRISMA 12-bit video records.

To account for count loss on the saturated portion of the Table 3. IT20200101 fireball parameters obtained from triangulation and dynamical model. The two columns refer to values at the beginning and end of the bright flight respectively (when applicable). Values of mass and diameter are computed from the mass-section ratio D (in the hypothesis of pure ablation) by assuming a spherical shape of the meteoroid and for the measured meteorite bulk density of 3.322 g cm -3 . PSF, a tentative correction is applied by fitting a bidimensional gaussian model to non-saturated pixels of the PSF. This analysis therefore aims to estimate the original PSF shape and unveils h values that are mostly below 2 (and always below 4), also for the two bright flares (Sect. 3.2). These conclusions are confirmed by a visual inspection of the saturated portion of the PSF, which is confined to few pixels in its very centre for most part of the trajectory. From this approach, a saturation correction factor is therefore estimated and applied to results obtained through aperture photometry. The computed count loss fraction for the saturated portion of the bright flight is mostly confined under 30 per cent and raises up to about 50 per cent in correspondence of the brightest flare.

Atmospheric trajectory and dynamical model

The results of the atmospheric trajectory and dynamical model computation are summarised in Tab. 3 and Fig. 4.

In particular, Fig. 4a shows the height of the fireball above the ground as a function of time since the beginning of the luminous path. The observed fireball trajectory begins at a starting height h b = 75.9 ± 0.2 km and ends up at a terminal height h t = 21.5 ± 0.1 km. The total length of the luminous atmospheric path is approximately 59 km, which was covered in about 5.6 s. The meteoroid followed an atmospheric trajectory inclined by an angle of 68. • 4 ± 0. is evident. Systematic effects associated to specific cameras may be related to PSF asymmetries that affect most part of the trajectory. The observed velocity (Fig. 4c) allowed us to derive an entering speed of 12.2 ± 0.2 km s -1 at 76 km of altitude. This speed started to decrease only in the denser layers of the atmosphere below 30 km, just after the brightenings, where it lowered to 10 km s -1 . At the end of the luminous path, 21.5 km above the ground, the meteoroid slowed down to 4.0 ± 0.2 km s -1 . The estimated value of 76 km for the beginning height of the observed trajectory is in close agreement with experimental and simulated data presented in Vida et al. (2018) for fireballs of asteroidal origin with a low entry speed (<13 km s -1 ), as seen by low sensitivity all-sky systems. They also found that low velocity meteors decelerate significantly prior to detection of the visible meteor trail. This is the case also for the Cavezzo bolide, for which we estimate a difference between v ∞ and the entry speed of 0.6 ± 0.3 km s -1 .

The absolute magnitude light curve is shown in Fig. 4e. In the first two seconds, the brightness grew rapidly, reaching a plateau of about M = -7.5 between t = 2.0 s and 4.7 s, followed by a sudden fading in the last second. Two rapid flares are visible at 3.95 s and 4.15 s with absolute magnitudes close to -9.5 and -8.5, respectively, related to the already mentioned fragmentation. The corresponding altitudes are about 32.6 km and 30.7 km, respectively, when the meteoroid was still moving at a speed of about 10.1 km s -1 . Meteoroid fragmentation models usually assume that the fragmentation process starts the aerodynamic pressure is equal to the mechanical strength S. According to the meteoroid height h and speed V in the main explosion (Foschini 1999), we can estimate a strength of about:

S = γ -1 γ ρ sl V 2 e -h/H ≃ 0.88 MPa , (1)
where γ ∼ 1.7 is the ratio of specific heats, H ≃ 8 km is the atmospheric scale height and ρ sl ≃ 1.22 kg m -3 is the atmospheric density at sea level. This value is close to the maximum aerodynamic pressure that is attained at a height of about 28.2 km (see Fig. 4d). Since Cavezzo is a stony meteorite, a strength of about 10 MPa could be expected, that is, about one order of magnitude greater. However, it is common to observe fragmentations at aerodynamic pressure well below 1 MPa and even down to few tens of kPa (for example, see Popova et al. 2011, Devillepoix et al. 2019). This evidence suggests a particular weakness of the meteoroid which may be caused, for instance, by fractures already present when it was entering the atmosphere or by porosity of the material.

We estimated the density of the F2 sample by an accurate and precise 3D scanning of the outer surface (Pratesi et al. 2020, submitted). Given the measured density of 3.322 g cm -3 and by assuming a spherical shape for the entering meteoroid, the computed value for the masssection ratio D derived by the dynamical model provides, in a purely ablative regime, an estimate of the mass and size before and after the luminous atmospheric transit. The meteoroid pre-atmospheric mass can be estimated to be 3.5 ± 0.8 kg (D = 280 ± 20 kg m -2 ) with a diameter of 0.13 ± 0.01 m, while the final mass is 1.5 ± 0.4 kg (D = 210 ± 20 kg m -2 ). Since there is evidence of possible fragmentation only at a height of about 30 km, we consider our estimation of the initial mass as reliable. On the contrary, the value of the final mass is questionable. Apart from this, the relatively low ablated mass ratio of about 57 per cent could be attributable also to the very low pre-atmospheric speed of 12.8 ± 0.2 km s -1 and to the steep inclination of the trajectory. Compared to the values given in Tab. 1, the preatmospheric mass for the Cavezzo meteorite is the lowest ever reported between "pedigree" meteorites. From the preatmospheric mass and velocity estimates, the impact energy results to be 0.07 ± 0.02 T TNT, which is also the lowest among values given in Tab. 1.

Atmospheric data

The knowledge of the atmospheric conditions plays a key role in the computation of the dark flight and strewn-field of meteorite fragments that could be possibly found on the ground. In particular, the wind direction and intensity are the major drivers for the loss of accuracy of these predictions. This effect is even more important in our case, given the small residual mass after ablation and the even smaller expected mass and size of the fragments (order of 100 g / 1 cm). For these reasons, we specifically computed the atmospheric state for the Cavezzo area at the time of fall.

Meteorological data came from IOIS (Integrated Observations Ingesting System) elaborated and used at Meteo Expert, a private organisation providing meteorological services where weather models are internally developed and applied. All available data, coming from surface, upper air and remote sensing measurements, are integrated to produce initialisation to be perturbed for a limited-area ensemble prediction system. A variational quality control is applied to check data consistency (Steinacker et al. 2011, Tavolato & Isaksen 2015); the observation is compared with the background and surrounding observations to determine its analysis weight in the system. This procedure is applied to develop perturbed initial data for usual forecast model, that runs at 00, 06, 12 and 18 UTC. However, in our case, as for several other applications including nowcasting, a hourly 3D grid is needed. This grid follows the modelâ ȂŹ s horizontal mesh size at 3.5 km, while in the vertical direction fifty variable-depth levels are used from surface to stratosphere. The scaling of weather parameters at a defined location is made by interpolating values from surrounding grid points, with a correction algorithm which takes into account subgrid terrain characteristics and local gradients.

Dark flight and strewn-field

Figure 5 shows the wind intensity and direction values at 18 UTC in the Cavezzo area as a function of the altitude. The wind was particularly intense at about 22 km, which is the last observed point of the luminous path, reaching a speed of about 28 m s -1 , and blowing at 45 • clockwise with respect to the meteoroid motion direction. The wind intensity decreases to reach a speed of about 20 m s -1 at 20 km altitude, and is confined below 10 m s -1 from 13 km downwards. This led to a significant shift of the strewn-field compared to a situation with zero wind, especially in the transverse direction. Figure 6 shows the computed strewn-field for the Cavezzo meteorite, together with the terminal part of the luminous atmospheric trajectory projected on the ground (red line), and the location where the two fragments were recovered (purple star). By assuming that a fragmentation occurred, the impact point computed with a purely ablative model is not representative of the real case. Therefore, we computed the expected impact point for fragments with different mass-section ratios, ranging from D = 30 up to 200 kg m -2 , which is approximately the final value for the mainmass pure ablation (Sect. 3.4). The brown thick line shows the most probable impact point as a function of D, while the red/orange/yellow shaded areas represent the 1/2/3σ uncertainty, respectively, in the transverse direction. The 1σ uncertainty in the longitudinal direction can be estimated to be 300 m. Table 4 reports the impact parameters for the sampled D values. The lateral displacement (X) can be as high as 2.3 km for smaller fragments. The expected impact velocity ranges from 28 to 71 m s -1 . The position of the recovered fragments lies at the very border of the 1σ transverse interval, in a region where recovered fragments are expected to have mass-section ratio values between 30 and 50 kg m -2 . The D values for the two fragments can be estimated to be 30 -70 kg m -2 for F1 and 35 -85 kg m -2 for F2, considering that we ignore the orientations of the meteorites during the fall. Therefore, the mass-section ratio for both F1 and F2 is compatible with the predicted values of the dark-flight model. However, since the meteorite very likely fragmented on the ground due to the impact, the D value of the original body remains unknown.

A POSSIBLE PROGENITOR OF THE CAVEZZO METEORITE

Using the value of 12.8 ± 0.2 km s -1 for the pre-atmospheric velocity, we computed the heliocentric orbit of the meteoroid prior to the impact. The true geocentric speed results to be 5.8 ± 0.5 km s -1 . The orbital elements are reported in Tab. 5. The apparent radiant (α a = 6. • 5 ± 0. we followed the procedure described in Carbognani et al. (2020), using the orbital similarity criterion D N introduced by Valsecchi et al. (1999); the NEODyS database10 conveniently lists the secular quantities used in D N for all Near-Earth Asteroids (NEAs) for which they are defined (Gronchi & Milani 2001). We use D N in the form:

D N = (U -U c ) 2 + (cos θ -cos θ c ) 2 + ∆ξ 2 , (2)
with:

∆ξ 2 = min ∆φ 2 I + ∆λ 2 I , ∆φ 2 I I + ∆λ 2 I I                            ∆φ I = 2 sin φ -φ c 2
∆φ I I = 2 sin 180 • + φ -φ c 2 ∆λ I = 2 sin λ ⊕ -λ ⊕,c 2
∆λ I I = 2 sin 180 • + λ ⊕ -λ ⊕,c 2 .
In the above expressions, U, θ, φ, λ ⊕ refer to the NEA, and are taken from NEODyS, while U c , θ c , φ c , λ ⊕,c are those of the Cavezzo meteorite (Tab. 5). Details on the definition of the variables and on D N are given in Valsecchi et al. (1999). We looked for NEAs for which D N ≤ 0.

(blue ellipse)

. There is a reasonable similarity between the two orbits, which is more evident if one considers the secular quantities that enter the computation of D N . On the other hand, the pair Cavezzo-2013 VC 10 is rather isolated in the 4-dimensional space U, θ, φ, λ ⊕ ; this isolation is recognisable even in the 2-dimensional space constituted by the ecliptical radiant coordinates. Figure 8 shows the radiants of the pair Cavezzo-2013 VC 10 , as well as the radiants of the simulated impactors of [START_REF] Chesley | Mitigation of Hazardous Comets and Asteroids[END_REF], and of the 20 real impactors listed in Granvik & Brown (2018). As discussed in the Appendix of Farnocchia et al. (2012), the radiant distribution simply reflects the values of the semi-major axis, eccentricity and inclination of those NEAs whose orbits actually can cross that of the Earth. As a result, NEA impactor radiants are not uniformly distributed in a plot like Fig. 8a, but present concentrations and regions of low density that are even more evident in Fig. 8b. It is in one of the low-density regions that the pair Cavezzo-2013 VC 10 is located, lending some additional credibility to the possible association.

CONCLUSIONS

After less than three years of operations, the PRISMA network, partner of FRIPON, has achieved one of its major objectives, i.e. the recovery of the first Italian meteorite by computation of a precise strewn-field through the analysis of its observational data. Two meteorite fragments, fallen near Cavezzo (Modena) on New Year's Day 2020, weighing 3.1 g and 52.2 g, were recovered three days after the fall as a result of a dedicated field search and thanks to the involvement of local people. They were found at a distance of 400 m from the nominal computed position, very close to the 1σ uncertainty value. These fragments were immediately recognised as freshly fallen meteorites, a fact that has been confirmed by the unquestionable presence of short-lived cosmogenic radioisotopes (such as 48 V, half-life of 15.97 d) measured with a very sensitive γ-ray detector at the INAF Monte dei Cappuccini Laboratory in Torino. The analyses carried out at the Department of Earth Sciences of the Firenze University highlighted strong differences between the two specimens and led the Nomenclature Committee of the Meteoritical Society to approve the classification proposal of anomalous L5 chondrite. The computed orbital parameters and the value of the Tisserand invariant (T J = 4.1 ± 0.2) are typical of a NEA with an aphelion located in the inner part of the main belt. The orbital elements, compared with the ones of known NEAs from the NEODyS database, show that only one object among those, namely 2013 VC 10 , is compatible with the Cavezzo meteoroid. Moreover, the radiants of both objects are located in a low-density region of NEA impactors, thus lending additional credibility to the association.

The associated fireball trajectory, observed by many eyewitnesses, was characterised by an entry velocity of 12.2 ± 0.2 km s -1 and by a high inclination angle of 68. • 4 ± 0. • 3. The luminous path started at a height of 75.9 ± 0.2 km and reached 21.5 ± 0.1 km, after travelling approximately 59 km in 5.6 s. The absolute magnitude reached a minimum of -9.5 at 32.6 km of altitude, where a bright flash was seen, very likely indicating a fragmentation and followed by a second flash reaching -8.5. In the brightest part of the trajectory, the mean absolute magnitude was around -7.5. By assuming a purely ablative regime, a spherical form of the meteoroid, and given the measured meteorite bulk density of 3.322 g cm -3 , the estimated residual mass is 1.5 ± 0.4 kg.

The pre-atmospheric mass and velocity of 3.5 ± 0.8 kg and 12.8 ± 0.2 km s -1 , respectively, leading to a total impact energy of less than 0.07 T TNT, together with the brightest absolute magnitude reached, are the lowest among those estimated for the 35 meteorites with "pedigree" recovered so far, as listed in Tab. 1. Currently, this recovery can be considered to date the most challenging in terms of size and magnitude of the associated event, proving the efficiency of the network and of our reduction pipeline even in such demanding conditions.

O R I G

  Cavezzo, the first Italian meteorite recovered by the PRISMA fireball network. Orbit, trajectory, and strewn-field D Gardiol, D Barghini, A Buzzoni, A Carbognani, M Di Carlo, M Di Martino, C Knapic, E Londero, G Pratesi, S Rasetti, et al. To cite this version: D Gardiol, D Barghini, A Buzzoni, A Carbognani, M Di Carlo, et al.. Cavezzo, the first Italian meteorite recovered by the PRISMA fireball network. Orbit, trajectory, and strewn-field. Monthly Notices of the Royal Astronomical Society, 2021, 501 (1), pp.1215-1227. 10.1093/mnras/staa3646.

F2F1Figure 1 .

 1 Figure 1. The two recovered samples of the Cavezzo meteorite. On the right, the first recovered fragment (F1, 3.1 g); on the left, the second and larger one (F2, 52.2 g).

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Figure 2 .

 2 Figure 2. Details of the larger fragment F2 of the Cavezzo meteorite. (a) White streaks occur on one edge of F2, suggesting on-ground breakup of the original body; (b) photomosaic of polarising optical microscope images (transmitted light, crossed polars) of a thin sections obtained from F2 (field width 12 mm) showing chondrules and chondrule fragments distributed in the matrix.

  d'Astrophysique de Marseille (LAM). Station monitoring, network security, software maintenance, real-time data processing and sharing tasks are in charge of FRIPON and LAM teams. PRISMA data are synchronised and stored at the IA2 (Italian Center for Astronomical Archives) INAF

Figure 3 .

 3 Figure3. Map of the PRISMA stations (white dots) involved in the detection of the IT20200101 fireball. The red line plots the fireball bright trajectory projected on the ground, and white circles enclose the fireball trail seen by each camera (reconstructed from video records). Please notice that fireball trails are oriented accordingly to the specific in-situ hardware installation, and may be not strictly consistent with one another (all-sky images, from which meteor trails are cropped, were approximately oriented with N direction upward and E direction rightward). Background map was generated using the Matplotlib Basemap Toolkit[START_REF] Hunter | [END_REF], https://matplotlib.org/basemap/users/index.html)

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  • 3 with respect to the horizontal plane, with an azimuth of 238. • 1 ± 0. • 2, travelling from WSW to ENE, and rapidly entered into the denser layers of the atmosphere. The height residuals of observed points with respect to the computed trajectory are plotted in Fig.4b. The standard deviation of σ h = 0.25 km, considering the distance of the fireball to the observing stations, corresponds to angles less then 10 ′ and it is therefore always below the pixel resolution of our cameras. Systematic patterns are visible along the trajectory. Above 50 km of altitude, the descent of the body into the atmosphere is slower in comparison to the fitted trajectory. Furthermore, after the two flare events, an increased spreading of the height residuals Downloaded from https://academic.oup.com/mnras/advance-article/doi/10.1093/mnras/staa3646/5998250 by University of Western Ontario user on 29

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15 .

 15 The et al. in only one candidate, 2013 VC 10 , for which proper elements are given in the last column of Tab. 5; for the pair Cavezzo-2013 VC 10 , D N = 0.115. The nominal orbit of 2013 VC 10 , projected onto the ecliptic plane, is shown in Fig. 7

Figure 4 .Figure 5 . 15 Figure 6 .

 45156 Figure 4. Results of trajectory computation and dynamical model for the IT20200101 fireball. (a) Vertical projection of the atmospheric trajectory; (b) vertical residuals of the atmospheric trajectory; (c) fireball velocity with respect to ground; (d) aerodynamic pressure (Eq. 1); (e) absolute magnitude (at 100 km, zenith of the observer). The x-axis reports the time elapsed from the beginning of the visible flight captured by the eight cameras. The two vertical dashed lines indicate the times at which the two flares occurred, 3.95 s and 4.15 s respectively. In every panel, grey points plot measured or computed values for the single stations, whereas the red line plots the nominal fit values (a-c) or a smoothed version of grey points (d,e). Red shaded area encloses 1σ uncertainty.

Figure 7 .Figure 8 .

 78 Figure 7. The reconstructed heliocentric orbit for the progenitor meteoroid of the Cavezzo meteorite (red ellipse) as seen from the ecliptic north pole and projected onto the ecliptic plane, together with the 1σ uncertainty band (shaded red area). The blue ellipse plots the 2013 VC 10 orbit (for which orbital elements are provided by the NEODyS database). Remaining ellipses plot Solar System planets' orbits up to Jupiter, and the black dots indicates their position along the orbit at the time of the IT20200101 fireball. The black small dots symbolically represent the asteroid Main Belt.

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Table 1 .

 1 List and relevant data of "pedigree" meteorites, i.e., for which recovery was accompanied by a sufficient set of sporadic or systematic observations (optical, radio, infrasound, seismic, satellite), allowing for pre-impact orbit reconstruction. From left to right: name of the meteorite (approved by The Meteoritical Society), date of fall, preatmospheric velocity and mass, estimated terminal mass, meteorite total known weight (TKW) recovered on the ground, minimum absolute magnitude recorded, impact energy (equivalent tons of TNT, 1 T = 4.187 • 10 9 J), fireball network which provided the observations (if any) and references for table data. The uncertainties associated to the values of the table are not given here for simplicity, but can be found in the respective references.

	Name	Date UT	v ∞ [km s -1 ] m ∞ [kg] m fin [kg] TKW [kg]	M a	E [T] b	Fireball Network c	Ref.
	P ÅŹ Ã bram	07/04/1959	20.9	1300	80	5.6	-19	70	CFN	1,2
	Lost City	04/01/1970	14.1	165	25	17	-12	4	PFN	3,4,5
	Innisfree	06/02/1977	14.7	42	4.9	4.58	-12.1	1	MORP	5,6
	Bene Å ֒ aov	07/05/1991	21.3	4100	300 d	0.0116	-19.5	200	EFN	5,7,8,9
	Peekskill	09/10/1992	14.7	5000	-	12.4	-16	130	-	2,10
	Tagish Lake	18/01/2000	15.8	56000	1300	10	-22	1700	-	11,12,13
	Mor à ֒ avka	06/05/2000	22.5	1500	100	1.4	-20	90	-	14,15,2
	Neuschwanstein	06/04/2002	20.9	300	20	6.22	-17.2	16	EFN	16,17,18
	Park Forest	27/03/2003	19.5	11000	-	30	-21.7	500	-	19,20
	Villalbeto de la Pe Ãśa	04/01/2004	16.9	600	13	5.2	-18	20	-	21,22,23
	Bunburra Rockhole	20/07/2007	13.4	22	1.1	0.339	-9.6	0.5	DFN	24,25
	Almahata Sitta	07/10/2008	12.4	40000	39	10.7	-19.7	730	-	26,27,28,29
	Buzzard Coulee	21/11/2008	18.0	10000	-	>200	-20	390	-	30,31,32
	Maribo	17/01/2009	28.3	2000	<20	0.0258	-20	190	-	33,34
	Jesenice	09/04/2009	13.8	170	20	3.611	-15	4	SFN	35,36
	Grimsby	26/09/2009	20.9	30	5	0.215	-14.8	2	SOMN	37
	Ko Å ֒ aice	28/02/2010	15.0	3500	500	11.3	-18	100	-	38,39
	Mason Gully	13/04/2010	14.5	40	-	0.0245	-9.4	1	DFN	40,41
	Kri Å¿evci	04/02/2011	18.2	50	<5 e	0.291	-13.7	2	CMN	42
	Sutterâ ȂŹ s Mill	22/04/2012	28.6	40000	-	0.943	-19	4000	-	43
	Novato	18/10/2012	13.7	80	-	0.363	-13.8	3	CAMS	44
	Chelyabinsk	15/02/2013	19.0	1.2•10 7	10000	730	-27.3	5•10 5	-	45,46,47
	Annama	18/04/2014	24.2	470	12.5	0.1679	-18.3	30	FFN	48,49,50
	Å¡d' Ã ֒ ar nad S Ã ֒ azavou	09/12/2014	21.9	150	>1.3 f	0.087	-15.3	9	EFN	51
	Porangaba	09/01/2015	-	-	-	0.970	-	-	-	52
	Sari Ãgi Ãgek	02/09/2015	17.3	1700	-	24.78	-16.8	60	-	53
	Creston	23/10/2015	16.0	50	-	0.8523	-12	2	CAMS, SACN	54
	Murrili	27/11/2015	13.7	38	2	1.68	-	0.9	DFN	55,56
	Ejby	06/02/2016	14.5	120	-	8.982	-14.0	3	-	57,58
	Stubenberg	06/03/2016	14	600	-	1.473	-15.5	14	EFN	59,60
	alov Ãl' Hradec Kr à ֒ Dishchiiâ ȂŹ bikoh	17/05/2016 02/06/2016	-16.6	-1000 g	--	0.134 0.07957	-11.5 -16	-30	EFN CAMS, SACN	61,62 63,64
	Dingle Dell	31/10/2016	15.4	40	1.4	1.150	-	1	DFN	65
	Hamburg	17/01/2018	15.8	140	>1	∼1	-16.3	5.5	-	66,67
	Renchen	10/07/2018	20	50 h	-	1.227	-13.4	2	EFN	61,68
	Cavezzo	01/01/2020	12.8	3.5	1.5	0.0553	-9.5	0.07	PRISMA	This work
	a Magnitude values are given in different passbands (e.g. visual, panchromatic) and might be not strictly comparable to one another b The impact
	energy was calculated by the authors, if not provided in the original work, or updated to more precise estimates of preatmospheric mass and/or
	velocity c CFN = Czechoslovakian Fireball Network (now EFN), PFN = Prairie Fireball Network, MORP = Meteorite Observation and Recovery
	Project, EFN = European Fireball Network, DFN = Desert Fireball Network, SFN = Slovakian Fireball Network (part of EFN), SOMN = Southern
	Ontario Meteor Network, CMN = Croatian Meteor Network (part of EFN), CAMS = Cameras for All-sky Meteor Surveillance, FFN = Finnish
	Fireball Network, SACN = Spalding Allsky Camera Network, SkySentinel d Most of the terminal mass in gram-sized meteorites e Apart from
	the main mass, just a few 10 -100 g meteorites are expected and ∼ 2000 meteorites with mass > 1 g f Main mass of 1.3 kg plus a second largest
	meteorite in the range 100 -200 g (∼ 250 meteorites in the range 10 -200 g, 6 kg total, and ∼ 3000 meteorites of 0 -1 g, 7 kg total) g There is a
	disagreement between meteoroid size deduced from radiated energy from satellite observations (∼ 15000 kg) and cosmogenic radionuclide data (400 -
	1800 kg) h Computed by the authors considering a preatmospheric radius of 10 -20 cm deduced from cosmogenic 26 Al data and a bulk density of
	3.4 g cm -3 (reference 68), assuming a spherical shape.

Table 2 .

 2 PRISMA stations that observed the IT20200101 fireball. From left to right: station name, latitude, longitude and elevation above sea level.

	Station name	Lat. N [ • ]	Long. E [ • ] El. [m]
	Bedonia (ITER04)	44 • 30 ′ 27. ′′ 7	09 • 37 ′ 57. ′′ 0	550
	Rovigo (ITVE02)	45 • 04 ′ 54. ′′ 0	11 • 47 ′ 42. ′′ 2	15
	Felizzano (ITPI03)	44 • 54 ′ 45. ′′ 0	08 • 26 ′ 14. ′′ 0	114
	Loiano (ITER01)	44 • 15 ′ 23. ′′ 7	11 • 19 ′ 54. ′′ 4	787
	Cecima (ITLO03)	44 • 48 ′ 52. ′′ 7	09 • 04 ′ 43. ′′ 6	670
	Navacchio (ITTO02)	43 • 40 ′ 59. ′′ 5	10 • 29 ′ 29. ′′ 9	15
	Padova (ITVE01)	45 • 24 ′ 07. ′′ 0	11 • 52 ′ 06. ′′ 7	64
	Asiago (ITVE03)	45 • 50 ′ 57. ′′ 9	11 • 34 ′ 06. ′′ 0	1370
	from visual observers (52 observations reported through the
	PRISMA website on the International Meteor Organization
	online form 2 ).

Table 4 .

 4 Data regarding the nominal impact points with different D final values. From top to bottom: final mass-section value, latitude and longitude of the impact point, shift parallel (L) and orthogonal (X) to motion direction of the bright flight and on-ground impact velocity.

	Quantity
	Final D [kg m 2 ]	30	40	50	60	70	85	100	125	150	175	200
	Lat. N impact point [ • ]	44.8245	44.8287	44.8318	44.8343	44.8364	44.8390	44.8412	44.8441	44.8466	44.8486	44.8504
	Long. E impact point [ • ] 10.9759	10.9773	10.9789	10.9804	10.9819	10.9841	10.9862	10.9893	10.9921	10.9947	10.9971
	L [km, ± 0.3]	0.4	0.7	1.0	1.2	1.5	1.8	2.0	2.4	2.7	3.0	3.3
	X [km, ± 0.4]	2.3	1.9	1.7	1.5	1.4	1.2	1.1	1.0	0.8	0.8	0.7
	v impact [m s -1 ]	28	32	36	39	42	47	51	56	62	66	71

  • 2, δ a = 30. • 6 ± 0. • 2) is located in the Andromeda constellation, while the true radiant (α t = 358. • 4 ± 0. • 3, δ t = 24. • 4 ± 0. • 3) is in Pegasus. The computed heliocentric orbit, shown as the red ellipse (nominal value) and shaded area (1σ uncertainty) in Fig.7, has moderate eccentricity and low inclination on the Ecliptic, with a Tisserand invariant with respect to Jupiter equal to 4.1 ± 0.2, thus indicating that the progenitor meteoroid was of asteroidal origin.To find possible progenitor(s) of the Cavezzo meteorite,

Table 5 .

 5 Orbital elements (top) and proper elements (bottom) of the Cavezzo meteoroid (left) and of 2013 VC 10 from NEODyS (right).

	Quantity	Cavezzo	2013 VC 10
	Epoch	J2000	MJD59000
	Semi major axis [AU]	1.82 ± 0.22	1.56622
	Eccentricity	0.460 ± 0.063	0.365295
	Inclination [ • ]	4.0 ± 1.6	2.044
	Long. of ascending node [ • ] 280.52311 ± 0.00001	224.068
	Argument of Perihelion [ • ]	179.2 ± 4.8	240.264
	Longitude of Perihelion [ • ]	99.7 ± 4.8	104.332
	Perihelion passage [JD]	2458849.6 ± 0.5	2458808.1
	Perihelion distance [AU]	0.983 ± 0.001	0.9941
	Aphelion distance [AU]	2.66 ± 0.41	2.1383
	U	0.216 ± 0.001	0.1818
	θ [ • ]	22.96 ± 0.30	24.8358
	φ [ • ]	175.90 ± 0.69	171.49
	λ ⊕ [ • ]	100.52311 ± 0.00001	104.986

https://prisma.imo.net/members/imo_view/event/2020/18

http://www.prisma.inaf.it/index.php/2020/01/02/una-meteorite-in-emilia-romagna

https://www.media.inaf.it/2020/01/02/forse-e-caduta-una-meteorite-in-emilia-romagna

https://www.lpi.usra.edu/meteor/metbull.php?code=72534

NUDAT databasehttps://www.nndc.bnl.gov/nudat2/ Downloaded from https://academic.oup.com/mnras/advance-article/doi/10.1093/mnras/staa3646/5998250 by University of Western Ontario user on 29 November 2020

IDL -Interactive Data Language -Harris Geospatial Solutions, Inc.https://www.harrisgeospatial.com/

MATLAB -Matrix Laboratory -MathWorks, Inc.,http://www.mathworks.com

NEODyS-2 databasehttps://newton.spacedys.com/~neodys2/propneo/en Downloaded from https://academic.oup.com/mnras/advance-article/doi/10.1093/mnras/staa3646/5998250 by University of Western Ontario user on 29 November 2020

Osservatorio Astronomico Bobhouse, Via Giuseppe Tomasi P.pe di Lampedusa 9, 90147 Palermo, PA, Italy This paper has been typeset from a T E X/L A T E X file prepared by the author. Downloaded from https://academic.oup.com/mnras/advance-article/doi/10.1093/mnras/staa3646/5998250 by University of Western Ontario user on 29 November 2020

ACKNOWLEDGEMENTS

PRISMA is the Italian network for systematic surveillance of meteors and atmosphere. It is a collaboration initiated and coordinated by the Italian National Institute for Astrophysics (INAF) that counts members among research institutes, universities, associations and schools. The complete list of PRISMA members is available here: http://www.prisma.inaf.it. PRISMA was funded by 2016/0476 and 2019/0672 Research and Education grants from Fondazione Cassa di Risparmio di Torino and by a 2016 grant from Fondazione Agostino De Mari (Savona). PRISMA data are hosted by the INAF research e-infrastructure project IA2 (Italian Center for Astronomical Archives). FRIPON was funded by the ANR grant N.13-BS05-0009-03, carried by the Paris Observatory, Mus Ãl'um National dâ ȂŹ Histoire Naturelle, Paris-Saclay University and Institut Pyth Ãl'as (LAM-CEREGE). FRIPON data are hosted and processed at Institut Pyth Ãl'as SIP (Service Informatique Pyth Ãl'as), and a mirror is hosted at IMCCE (Institut de M Ãl'canique C Ãl'leste et de Calcul des ÃL'ph Ãl'm Ãl'rides / Paris Observatory) with the help of IDOC (Integrated Data and Operation Center), supported by CNRS and CNES.

The PRISMA collaboration is grateful to Mr. Davide Gaddi, who recovered the meteorite fragments, allowed for the analysis of the samples and eventually decided to donate them to INAF.

The authors are very grateful to Leonard Korno Å ֒ a for the useful comments that greatly improved the manuscript and to Rebecca O'Brien for the precious work of language revision.

Background map data for Fig. 6 copyrighted OpenStreetMap contributors and available from https://www.openstreetmap.org, distributed under the Open Data Commons Open Database License (https://opendatacommons.org/licenses/odbl/). Downloaded from https://academic.oup.com/mnras/advance-article/doi/10.1093/mnras/staa3646/5998250 by University of Western Ontario user on 29 November 2020

DATA AVAILABILITY

PRISMA data underlying this article will be shared on reasonable request to the corresponding author.