Monica Capone
email: monica.capone@unito.it

Matteo Luca Ruggiero
email: matteo.ruggiero@polito.it

Jumping from Metric f (R) to Scalar-Tensor Theories and the relations between post-Newtonian parameters

We review the dynamical equivalence between f (R) gravity in the metric formalism and scalartensor gravity, and use this equivalence to deduce the post-Newtonian parameters γ and β for a f (R) theory, obtaining a result that is different with respect to that known in the literature. Then, we obtain explicit expressions of these paremeters in terms of the mass of the scalar field (or, differently speaking, the mass of the additional scalar degree of freedom associated to a f (R) theory) which can be used to constrain f (R) gravity by means of current observations.

I. INTRODUCTION

In f (R) theories of gravity the gravitational Lagrangian depends on an arbitrary analytic function, f , of the scalar curvature, R. The field equations of these theories (see [1], [2] and references therein) can be obtained in the metric formalism, by varying the action with respect to the metric tensor, or in the Palatini formalism, where the action is varied with respect to the metric and to the affine connection, which are supposed to be independent from each other. In both cases the matter Lagrangian is supposed to depend only on the matter fields and the metric tensor. In addition to these two formalism, there is also the metric-affine formalism, in which one performs the Palatini-type variation supposing the matter Lagrangian to depend upon the metric tensor and the affine connection [3], besides the matter fields. These theories are also referred to as higher-order theories of gravity, since the field equations are of higher order with respect to those of General Relativity (GR).

The dynamical equivalence between f (R) theories and a particular class of scalar-tensor (ST) theories, i.e. Brans-Dicke (BD) theory is very well-known and studied, both in the case of metric formalism [4], [5], [6] as well as in the Palatini formalism [7], [8]. We stress that two theories are said to be dynamically equivalent (limiting ourselves to the classical point of view) when it is possible to make their field equations (or actions) coincide by means of suitable redefinitions of the fields (gravitational and matter). In particular, the dynamical equivalence between f (R) and the BD theory (see e.g. [2] and references therein) suggests to use the results known for the latter to directly obtain, after suitable manipulations, those corresponding to the former. This could be very fruitful especially for those results directly related to observations or experiments: for instance, post-Newtonian parameters (see [9] for the original works, [START_REF] Will | Theory and Experiment in Gravitational Physics[END_REF] for a detailed description of the framework and [START_REF] Will | [END_REF] for a recent review) can be used to constrain f (R) theories. The issue of the post-Newtonian parameters for metric f (R) theories has been previously dealt with in the literature, following the approach described in [12]. However, the choice of the starting parameters and a too peculiar selection for the transformation between f (R) and the corresponding BD theory have led to at least misleading results.

In this paper, after reviewing the dynamical equivalence between f (R) and ST theories, we focus on the post-Newtonian parameters issue. In particular, in the framework of metric f (R) gravity, we discuss how post-Newtonian parameters are obtained, by exploiting the dynamical equivalence with ST theories, and we suggest that those obtained in [12] are not correct. Then we deduce the post-Newtonian parameters γ and β for a metric f (R) theory and, furthermore, we give the expressions of these two parameters in terms of the mass of the scalar field, both for a general scalar-tensor theory and for a f (R) theory.

II. GENERAL CORRESPONDENCES BETWEEN f (R) GRAVITY AND SCALAR-TENSOR THEORIES

We discuss the dynamical equivalence between f (R) and ST theories, focusing on the simplest case of a fourth-order Lagrangian (see [6], [13], [14] for the extension to theories of order higher than the fourth, i.e., of the type f (R, n R), with n ≥ 1).

By varying the action integral of f (R) gravity1

S HO = 1 16πG N d 4 x √ -gf (R) + S m , (1)
where

S m = d 4 x √ -gL m (2)
is the matter action and L m is the matter Lagrangian, with respect to the components of the metric tensor, the following fourth-order equations of motion are obtained [15],

G µν = 1 f (R) 1 2 g µν [f (R) -Rf (R)] + f (R) ,µ;ν -g µν f (R) + 8πG N f (R) T µν (3)
where G µν represents the Einstein tensor and

T µν . = - 2 √ -g δL m δg µν (4)
are the components of the matter/energy fields stress-energy tensor. Now, let us consider the scalar-tensor action integral

S φ = 1 16πG N d 4 x √ -g [f (φ) + (R -φ)f (φ)] + S m . (5)
By varying it with respect to the metric tensor, we get the equations of motion

G µν = 1 f (φ) 1 2 g µν [f (φ) -φf (φ)] + f (φ) ,µ;ν -g µν f (φ) + 8πG N f (φ) T µν , (6)
whereas a variation with respect to the scalar field, φ, gives the equation

[R -φ] f (φ) = 0. ( 7
)
As a consequence, it is easy to realize that, for a given function f , provided that f (φ) = 0, Eqs.(6) coincide with Eqs.(3) on shell, i.e. on the solutions of the last equation (7), so when

R = φ. (8)
This correspondence on shell can be demonstrated to apply also in N dimensions and with a more general higherorder Lagrangian, i.e. for (2n + 4) th -order gravity [6], [13], [14] (see also [16] for a representation of the higher-order Lagrangian density when also negative powers of the Dalambertian operator are present). Now, it is interesting to point out that the action (5) can be transformed into a general ST action,

S ST = 1 16πG N d 4 x √ -g [F (ϕ)R -Z(ϕ)g µν ∇ µ ϕ∇ ν ϕ -2V (ϕ)] + S m , (9)
by setting in the latter the identifications: Consequently, by varying the action ( 9) with respect to both the components of the metric and the scalar field we obtain the following equations of motion,

F (ϕ) = f (φ), Z(ϕ) = 0, 2V (ϕ) = φf (φ) -f (φ). ( 10
) Type Tricks L/ √ -g GST F (ϕ)R -Z(ϕ)g µν ∇µϕ∇ν ϕ -2V (ϕ) GBD F (ϕ) ≡ ϕ, Z(ϕ) ≡ ω/ϕ ϕR -ω ϕ g µν ∇µϕ∇ν ϕ -2V (ϕ) O'H ω = 0 ϕR -2V (ϕ) ST ϕ ≡ f (φ), 2V (ϕ) ≡ φf (φ) -f (φ) f (φ) + (R -φ)f (φ) HOT φ ≡ R f (R)
G µν = 1 F (ϕ) Z(ϕ) ∇ µ ϕ∇ ν ϕ - 1 2 g µν ∇ σ ϕ∇ σ ϕ + F (ϕ) ,µ;ν -g µν F (ϕ) -g µν V (ϕ) + 8πG N F (ϕ) T µν , (11)
and

ϕ + 1 2Z(ϕ) R dF dϕ + ∇ σ ϕ∇ σ ϕ dZ dϕ -2 dV dϕ = 0, (12)
respectively.

As it is well-known [17], the action integral ( 9) reduces to the generalized Brans-Dicke one by simply substituting

F (ϕ) = ϕ, Z(ϕ) = ω ϕ . (13)
So, on taking into account Eqs ( 10) and ( 13), we see that f (R) theories can be suitably transformed into a BD theory with ω = 0.

In table I a synopsis of the key steps for going to a generalized scalar-tensor Lagrangian to an f (R) one are described.

Remark.

According to what has been stated above, metric f (R) gravity is nothing but a different representation of the Brans-Dicke theory with null BD parameter, ω = 0: as a consequence, metric f (R) gravity has one extra degree of freedom with respect to General Relativity. Actually, this extra degree of freedom is dynamic, as one can easily deduce from the equation of motion for the ϕ field, obtained from Eq.( 12) with the substitutions (13), in which we set ω = 0, after the replacement of the Ricci scalar with the trace of Eq. [START_REF] Will | [END_REF], that is

3 ϕ + 4V (ϕ) -2ϕ dV (ϕ) dϕ = 8πG N T. (14)
Of course, one should also bear in mind that the shape of the potential V (ϕ) of the particular BD theory we are referring to is constrained by the scalar curvature. Precisely, it is

2V (ϕ) = R, (15)
as one immediately deduces from the relations ( 8) and (13).

III. POST-NEWTONIAN PARAMETERS IN METRIC f (R) GRAVITY

We now focus on the deduction of the post-Newtonian parameters for metric f (R) gravity.

To begin with, we consider the approach outlined in [18], where the post-Newtonian parameters for a general scalartensor theory of gravity are obtained, provided that we can disregard the potential associated to the scalar field. On doing so, we exploit the correspondence between f (R) and scalar-tensor gravity discussed above by considering now what happens when we conformally transform a scalar-tensor and a higher-order gravity Lagrangian. It can be shown [5], [19] that both theories are mapped into General Relativity plus a suitable number of scalar fields: more precisely, one can demonstrate that a scalar-tensor theory action is conformally equivalent to the Einstein-Hilbert action plus as many scalar fields as there are in the former action. On the other hand, when a conformal transformation of an f (R) action from the starting frame (referred to as "Jordan frame") to the final one (called "Einstein frame") is performed, one finds that it is equivalent to Einstein-Hilbert plus a scalar field action.

Before going on, it is useful to point out that there is a longstanding debate on the issue of conformal transformations, between those who try to establish which frame is the physical one (see for example [20], [21] and references therein for a general discussion) and those who maintain that the two frames, under suitable redefinitions of the units, are physically indistinguishable (see [22], [23]). A thorough discussion on this issue is beyond the scope of the present paper, however the reader can refer to the huge literature on this interesting problem. Now, after performing the conformal transformation

           gµν ≡ Ω 2 (ϕ)g µν = F (ϕ)g µν , dψ dϕ 2 ≡ 3 4 d ln F (ϕ) dϕ 2 + Z(ϕ) 2F (ϕ) , A(ψ) ≡ Ω -1 (ϕ) = F -1/2 (ϕ), 2U (ψ) ≡ V (ϕ)F -2 (ϕ), (16)
the action integral ( 9) takes the form of the Einstein-Hilbert action plus a scalar field [18], [17],

S Conf T r = 1 4πG N d 4 x √ -g R 4 - 1 2 gµν ∇ µ ψ∇ ν ψ -U (ψ) + S m Ψ m ; A 2 (ψ)g µν , (17)
where the matter term is now non-minimally coupled to the scalar field ψ through the conformal factor A2 (ψ). Note that the new dynamical degree of freedom ψ corresponds to the starting dynamical degree of freedom, i.e. to the scalar field ϕ. On the other hand, when we conformally transform the action integral (1) using the conformal factor

Ω 2 (ϕ) = df dR , (18)
the conformal transformation at stake turns out to be defined by the relations

           gµν ≡ Ω 2 (ϕ)g µν = f (R)g µν , ψ = 3 2 1 8πGN ln (f (R)) , A(ψ) ≡ Ω -1 (ϕ) = [f (R)] -1/2 , 2U (ψ) ≡ 1 8πGN Rf (R)-f (R) f 2 (R) , (19)
where the prime indicates the derivative with respect to the scalar curvature, R. The scalar field, ψ, that materializes in the Einstein frame corresponds to the dynamical degree of freedom, ϕ, of the BD representation one can associate to f (R) gravity. Eventually, by varying such a transformed action, we are left with the subsequent set of equations

Gµν = 2∂ µ ψ∂ ν ψ -gµν ∂ σ ψ∂ σ ψ -2g µν U (ψ) + 8πG N Tµν , (20)
for the components of the metric tensor and

˜ ψ = -4πG N α(ψ) T + dU (ψ) dψ (21)
for the scalar field, where

α(ψ) ≡ d ln A(ψ) dψ (22)
is a function giving the strength of the coupling between the scalar field and the matter/energy source, while T is the trace of the stress-energy tensor 2 back. It is worth stressing that Eqs. (20) and Eqs.( 21) are exactly those one would obtain by transforming the Eqs.( 11) and (12), respectively.

The procedure and results shown above are generalizable to actions of order greater than the fourth. In this case, the Lagrangian (that is function of R, R, ..., n R) is dynamically equivalent to a BD Lagrangian given in terms of (n + 1) scalar fields and with BD parameters all equal to zero. This last Lagrangian is conformally equivalent to General Relativity plus (n + 1) scalar fields 3 .

Having this in mind, we can proceed to determine the post-Newtonian parameters γ and β for a metric f (R) gravity. Actually, according to the procedure described in [18], for a scalar-tensor action integral whose potential term can be neglected, these parameters turn out to have the following expressions:

γ -1 = -2 α 2 1 + α 2 ψ0 , (23)
and

β -1 = 1 2 α 2 (1 + α 2 ) 2 dα dψ ψ0 , (24)
where ψ 0 is the asymptotic value of the field ψ. Taking into account the relations ( 16), we can express them back into the Jordan frame. After straightforward algebra, they read:

γ -1 = - (dF/dϕ) 2 ZF + 2(dF/dϕ) 2 ϕ0 , (25)
β -1 = 1 4 F (dF/dϕ) 2ZF + 3(dF/dϕ) 2 dγ dϕ ϕ0 , (26)
where now we have to consider the asymptotic value of the field ϕ. These two relations, written for a general Brans-Dicke theory, give the well-known results

γ = 1 + ω 2 + ω , (27)
and

β -1 = 0, (28)
already obtained in [START_REF] Will | [END_REF] without passing through a conformal transformation.

In order to deduce the values of the post-Newtonian parameters γ and β for a metric f (R) gravity, we start by pointing out what follows: in general, the above parameters ( 23)- (26) are not suitable for our purposes. In fact, they have been obtained by requiring the potential associated to the scalar field of the theory to be absolutely negligible. This is not true a priori: in fact, the potential we are now considering is strictly connected to the choice of the particular f (R), as it is

2V (ϕ) = Rf (R) -f (R). (29)
Actually, this fact has not been considered in [12], where the authors start from the correspondence between metric f (R) and scalar-tensor theories, namely from the relations ( 23)-( 26) above, to obtain the following expressions

γ R -1 = - f (R) 2 f (R) + 2f (R) 2 R0 , (30)
and

β R -1 = 1 4 f (R)f (R) 2f (R) + 3f (R) 2 dγ R dφ R0,φ0 , (31)
being R 0 , φ 0 the asymptotic values of the scalar curvature and the φ field, respectively. Apart from the improper use of the relations ( 23)-( 26), the above results ( 30) and (31), are not in agreement with those one would obtain by making use of the substitutions that led us from a higher-order theory to the corresponding ST one, which have bene discussed above. In fact, by setting F (ϕ) ≡ ϕ ≡ f (φ) and φ ≡ R, one obtains the following expressions for the two considered post-Newtonian parameters:

γ -1 = - [f (R) dR dϕ ] 2 Zf (R) + 2[f (R) dR dϕ ] 2 ϕ0 , (32)
and

β -1 = 1 4 f (R)f (R) dR dϕ 2Zf (R) + 3[f (R) dR dϕ ] 2 dγ dR dR dϕ ϕ0 (33) = 1 4          f (R)f (R) dR dϕ 2Zf (R) + 3 f (R) dR dϕ 2 Zf (R) (f (R)) 2 -2f (R)f (R) Zf (R) + 2 f (R) dR dϕ 2 2          ϕ0 . ( 34
)
These expressions for γ and β can be made equal to γ R and β R , respectively, only for a specific and peculiar choice of the fields ϕ and Z(ϕ). In fact, the equivalence between the two sets of relations, ( 30)-( 31) and ( 32)-( 34), only takes place when Z = 1 and dR = dϕ. In particular, the position Z = 1 implies f (R) = ω whereas dR = dϕ implies f (R) = 1. Even allowing ω to be a function of the curvature and not a constant parameter, putting together the two positions, leads us to the conclusion that the BD parameter and the f (R) are obliged to be ω = R and

f (R) = (R 2 /2) + R, respectively 4 .
Let us further comment on the issue of the interpretation of the post-Newtonian parameters ( 23) and (24). Suppose to work with a class of higher-order theory such as f (R) = R 1+ , with → 0. What one would expect is that, as far as → 0, the post-Newtonian parameters reach their GR values (i.e. γ = 1, β = 1). On the other side, the relations ( 27) and ( 28), with ω = 0 (implying that it is also Z(ϕ) = 0), immediately give the results γ = 1/2 and β = 1. The key point to solve this puzzle is the role of the Z function. In fact, we must consider in our discussion that the more we approach GR, the closer Z approaches the infinity, and, consequently, γ → 1 as → 0. This further shows that a superficial application of the correspondence between f (R) and scalar-tensor theories leads to results that are non self consistent.

In next section, we are going to focus on the issue of the relevance of the potential associated to the scalar field when one wants to obtain the values of the post-Newtonian parameters γ and β for metric f (R) theories starting from the correspondence with scalar-tensor gravity; as we are going to show, this fact can be related to the mass of the scalar field.

IV. POST-NEWTONIAN PARAMETERS AND THE MASS OF THE SCALAR FIELD

As pointed out in [2] and [24], the ranges (or, equivalently, the masses) of the Brans-Dicke scalar field ϕ can be completely different according to which definition of the mass is used. Of course, this choice turns out to be crucial if we aim at writing the post-Newtonian parameters γ and β in terms of the mass of the scalar field 5 . So, before going on, it is useful to clarify this issue, in order to specify which definition of mass we refer to in our approach.

To begin with, we notice that we can assign to a scalar field endowed with a potential V (ϕ) the mass

µ 2 (ϕ) ≡ d 2 V dϕ 2 , (35)
that derives from the usual Klein-Gordon equation for a scalar field ϕ, that is

ϕ - d 2 V dϕ 2 = S, (36)
where S represents the source term. Note that, with this definition, the mass turns out to be dimensionless, having both V and ϕ 2 the dimensions of the fourth power of a mass6 . Actually, the field ϕ that appears in the action integral of the general Brans-Dicke theory,

S BD = 1 16πG N d 4 x √ -g ϕR - ω ϕ g µν ∇ µ ϕ∇ ν ϕ -2V (ϕ) + S m , (37)
satisfies a modified Klein-Gordon equation,

ϕ = 1 3 + 2ω [8πGT + 2ϕV (ϕ) -4V (ϕ)] , ( 38
)
where T is the trace of the matter/energy stress-energy tensor and the prime stands for derivative with respect to the ϕ field. Eq.( 38) can be immediately recast into a Klein-Gordon one, by the introduction of an effective potential V ef f (ϕ) such that

dV ef f (ϕ) dϕ = 2 3 + 2ω [ϕV (ϕ) -2V (ϕ)] , (39)
so that Eq.( 38) becomes

ϕ - dV ef f (ϕ) dϕ = 8πGT 3 + 2ω . ( 40
)
Then, an effective mass, m(ϕ), can be introduced as follows:

m 2 (ϕ) . = d 2 V ef f dϕ 2 = 2 3 + 2ω (ϕV -V ). ( 41
)
We notice that, in this case, the mass has the proper dimensions, as now V ef f has the dimensions of the sixth power of a mass. Thus, the correct use of the Klein-Gordon analogy seems to support the definition (41) rather than (35).

For the sake of completeness, we remember a third possible definition of mass, the one in the Einstein frame,

m2 (ψ) ≡ d 2 U dψ 2 . (42)
We also stress that, in this case, there are in principle as many definitions as the possible different choices of the conformal factor. It is useful to point out that these three different definitions do describe different physical situations, as it appears clear when considering the simple example of a constant potential V = V 0 = 0. In this case, it is µ = 0 = m(ϕ) thus giving an infinite range scalar field, but m = 0, giving a finite range scalar field.

That being said, we see that the range of ϕ is completely determined by the equation of motion (40) which, in the weak-field, slow-motion and spherically symmetric limit (suitable to describe the situation inside our low-density Solar System), becomes

1 r 2 d dr r 2 dϕ(r) dr - dV ef f (ϕ) dϕ 0. (43)
Now, the effective mass is obtained by expanding the potential term around the present value of ϕ, ϕ 0 , as follows:

dV ef f (ϕ) dϕ dV ef f (ϕ) dϕ     0 + d 2 V ef f (ϕ) dϕ 2     0 ϕ = m 2 (ϕ 0 )ϕ, (44)
where the last equality has been attained supposing V ef f to have a minimum at ϕ 0 . Equation (43) admits the usual Yukawa-like solution ϕ(r) ∝ exp[-m(ϕ 0 )r]/r with the mass determined by the definition (41). It then appears clear that the right definition of the mass (and, consequently, the proper range for the scalar field) to be singled out and consequently used in the analysis of post-Newtonian parameters is Eq.(41) [24].

Eventually, note that from the same definition, on taking into account Eqs. ( 15), (29), one gets the mass for the case of a generic f (R) theory,

m 2 HO = f -Rf 3f , (45)
provided that f = 0, where primes mean derivative with respect to R. This result is in agreement with that found in [24] as well as in other contests, like studies of stability [26], perturbations [27] and propagator calculations for f (R) [28], etc. [29].

For example, if we consider a Lagrangian that can be expressed as power series of the Ricci scalar around the asymptotic value R = 0, we may write, in general

f (R) = c 0 + c 1 R + c 2 R 2 + ... + c n R n (46)
where c i are constant coefficients. Then, from (45), after evaluating around R = 0, we obtain

m 2 HO = c 1 6c 2 (47)
so that the mass is directly related to the parameters of the Lagrangian.

We are now able to write the post-Newtonian parameters in terms of the effective mass (41) of the scalar field, ϕ, and the shape of the potential, V (ϕ). In fact, as we pointed out above, this is a crucial point that should be considered in order to translate the PPN parameters from a scalar-tensor to an f (R) theory. As effectively stated in [30] and in [31], for a massive scalar field like the case we are discussing (where none of the two extreme conditions m(ϕ)r 1 and m(ϕ)r 1, with r that indicates the scale of the experiment or observation testing the dynamics of the field, is in principle required), the correct expression of γ is:

γ = 3 + 2ω -e -m0r 3 + 2ω + e -m0r , (48)
where m 0 = m(ϕ 0 ) is the background value of the mass, that is

m 2 0 = 2 3 + 2ω (ϕV -V )   0 . ( 49
)
Remark. The expression (48) should be interpreted as an effective value of the post-Newtonian parameter γ, because of the dependence on the scale of the experiment r; as shown in [30] and [31], this is due to a Yukawa-like correction to the Newtonian potential 7 . In particular, it can be used to exploit the known bounds on γ to constrain the allowed values of m 0 and ω, on the experimental scale r such that m 0 r 1, so that these constraints are scaledependent 8 . As for the f (R) case, this fact implies that it is possible to constrain the parameters of the Lagrangian thanks to relations similar to Eq. (47). For the sake of completeness, we notice that when the background value of the mass of the scalar field is small, m 0 1/r (i.e., light, long range field), the value of the parameter γ, on which the observational bounds are directly applicable, is practically independent of the mass itself. In this case, the constraints on γ can be turned into a constraints on ω and the relation (48) reduces to the well-know one [START_REF] Will | [END_REF],

γ = 1 + ω 2 + ω . ( 50
)
On the other hand, when the mass is very large, m 0 1/r (i.e., very massive, short range field), the γ parameter is inexorably driven to its GR value, that is one.

TABLE I :

 I In this table, the steps to follow in order to pass from a generalized scalar-tensor (GST) Lagrangian density, normalized to the square root of the metric determinant, to an f (R) one are indicated. We have indicated, respectively, with GBD the generalized Brans-Dicke, O'H the O'Hanlon, ST the scalar-tensor and HOT the higher-order type normalized Lagrangian density.

We use units such that c = 1.

From the very definition of the stress-energy tensor also valid in the Einstein frame, Tµν ≡ 2 (δSm/δg µν )/ √ -g, it is easy to deduce its relation with its Jordan frame counterpart, Tµν = A 2 (ψ)Tµν . As a consequence, the contracted Bianchi identities give us the tensorial relation ∇µ T µ ν = α(ψ) T ∇ν ψ.

To be more precise, we can in general define only one scalar field in the Einstein frame which turns out to have a standard kinetic term, while the remaining degrees of freedom in Einstein frame will have non-standard kinetic terms and may also be coupled via the potential[6].

Of course, in this case, the two Eddington PPN parameters γ and β, being constants, must be evaluated in correspondence with the asymptotic value R 9

Historically, the interest in the study of scalar fields provided with a mass such that the experimental limits on the ω parameter were bypassed, started after that (bosonic) string theory showed to have a low-energy limit corresponding to ω = -1 Brans-Dicke theory[25]

Besides c = 1, we also set = 1.

It is interesting to point out that Yukawa-like corrections arise in f (R) gravity also without making use of the analogy with scalar-tensor gravity (see e.g.[32,33]).

We point out that our approach cannot be applied in principle to experiments that develop at very different scales, such as gravitational lensing; in these cases, a generalized parametrized post-Newtonian formalism should be applied that takes into account extra terms arising in f (R) gravity (see e.g.[32]).

Acknowledgments

The authors would like to thank two anonymous referees for their suggestions which contributed to improve the paper. MC is supported by Regione Piemonte and Università degli Studi di Torino.

We can rewrite the ω parameter in terms the effective mass of the field, Eq.( 41),

and substituting it into the relation (48) gives

We remark that this equation is only valid in the intermediate range, that is when the mass of the scalar field is comparable to the range, m 0 1/r. As for the β parameter, it can be shown [31], [30] that it is not affected by the presence of a nonnull mass for the scalar field, thus remaining frozen to the value it gets in the case of massless BD theories (as well as in GR), i.e., it is β = 1.

At this point, we can finally work out the post-Newtonian parameters for a metric f (R) theory of gravity. On taking into account Eqs. ( 45) and (48), on setting ω = 0 according to what we have seen in Section II, we obtain

and of course

where m 0 HO is the background value of the mass associated to a f (R) theory, given by (45), in agreement with [31]. For instance, on taking into account the solar system constraints on γ from the Cassini mission [34],

it is possible to constrain m 0 HO . As an example, since in this case the scale is r = 1 AU 1.5 × 10 8 km, that is to say a mass scale m AU 10 -27 GeV, we obtain m 0 HO mAU 10 which, in turn, can be used to constrain the parameters of the Lagrangian by means of Eq. (45).

V. CONCLUSIONS

We reviewed the issue of the dynamical equivalence between f (R) and scalar-tensor theories, focusing on the metric formulation of f (R) theory, in order to point out how the field equations for scalar-tensor theories reduce to f (R) field equations. On exploiting this equivalence, we gave the expressions of the post-Newtonian parameters γ and β for a general f (R) theory, which are not the same as those available in the literature, also because the latter have been obtained by using relations that implicitly require the potential of the scalar field to be negligible. Furthermore, we gave explicit expressions of these paremeters in terms of the mass of the scalar field, or differently speaking, the mass of the additional scalar degree of freedom associated to an f (R) theory. These expressions could be used to constrain f (R) theories by means of the values of the post-Newtonian parameters obtained by the available observations and tests, as we have shown by means of a simple example. Eventually, we showed that if the mass of the scalar field is very light, values of these parameters that are in agreement with observations are obtained; on the other hand, if the scalar field is heavy, the scalar field is essentially suppressed, and the predictions of the theory are undistinguishable from those of General Relativity.


Marie Chavent
email: chavent@math.u-bordeaux1.fr

Stéphane Girard
email: stephane.girard@inrialpes.fr

Vanessa Kuentz
email: vanessa.kuentz@cemagref.fr

Benoit Liquet
email: benoit.liquet@isped.u-bordeaux2.fr

Thi Mong

Ngoc Nguyen
email: nguyen@math.u-bordeaux1.fr

Jérôme Saracco
email: saracco@math.u-bordeaux1.fr

Benoît Liquet

An adaptive SIR method for block-wise evolving data streams

Keywords: dimension reduction, sliced inverse regression (SIR), data stream

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In the framework of high dimensional data, we consider the following semiparametric dimension reduction single index model proposed by Duan and Li [START_REF] Duan | Slicing regression: a link-free regression method[END_REF]:

Y = f (X β, ) (1)
where the univariate response variable Y is linked with the p-dimensional regressor X (with expectation E(X) = µ and covariance matrix V (X) = Σ) only through the single index X β. The error term is independent of X.

The link function f and the vector β are unknown. Since β is not totally identifiable in this model, we are interested in finding the linear subspace spanned by β, called the Effective Dimension Reduction (EDR) space.

In this paper we focus on data arriving sequentially by block in a stream. We assume that each data block t is composed of an independent and identically distributed (i.i.d.) sample {(X i , Y i ), i = 1, . . . , n t } available from model [START_REF] Bernard-Michel | Retrieval of Mars surface physical properties from OMEGA hyperspectral images using Regularized Sliced Inverse Regression[END_REF]. Our goal is to estimate the EDR direction at each arrival of a new block of observations. A simple direct approach consists in pooling all the observed blocks and then estimate the EDR direction by the Sliced Inverse Regression method introduced by Li [START_REF] Li | Sliced inverse regression for dimension reduction (with discussion)[END_REF]. While SIR is a computationally simple and fast method, the drawback of pooling the data is the storage of the blocks since the size of the dataset considerably increases with the number of blocks. To avoid this, we propose an adaptive SIR method based on the SIR approach for a stratified population developped by Chavent et al. [START_REF] Chavent | A sliced inverse regression approach for a stratified population[END_REF]. The proposed adaptive SIR method will be used to evaluate the physical properties of surface materials on the planet Mars from hyperspectral images. Our goal is to estimate the function G between some physical parameters Y and observed spectra X. To this end, a stream of synthetic spectra is generated by a physical radiative transfer model. The high dimension of spectra (p = 184 wavelengths) will be reduced using regularizd SIR (see Bernard-Michel et al. [START_REF] Bernard-Michel | Retrieval of Mars surface physical properties from OMEGA hyperspectral images using Regularized Sliced Inverse Regression[END_REF] for details) and the proposed adaptive SIR method.

An adaptive SIR estimator

Recall on SIR in block t. The population version SIR relies on the following linear condition:

(C) : ∀b ∈ p , E(X b|X β) is linear in X β,
which is fulfilled when X is elliptically distributed. Moreover, in the presence of high-dimensional data, this condition is often approximately fulfilled, see Hall and Li [START_REF] Hall | On almost linearity of low dimensional projections from high dimensional data[END_REF] for details. Let us consider a monotone transformation T (.) of Y . Under condition (C) and model (1), Li [START_REF] Li | Sliced inverse regression for dimension reduction (with discussion)[END_REF] showed that the centered inverse regression curve is contained in the one-dimensional linear subspace of p spanned by Σβ. As a consequence, the eigenvector b t of Σ -1 Γ t associated with the nonnull eigenvalue is an EDR direction (i.e. is collinear with β) where Γ t = V (E(X|T (Y ))).

To obtain an estimator of Γ t which can be easily used in practice, Li [START_REF] Li | Sliced inverse regression for dimension reduction (with discussion)[END_REF] proposed for T (.) a slicing into H t ≥ 2 non-overlapping slices s 1 , . . . , s Ht . Denoting the hth slice weight (resp. mean) by p h = P (Y ∈ s h ) (resp. m h = E(X|Y ∈ s h )), then the matrix Γ t can be written as

Γ t = Ht h=1 p h (m h -µ)(m h -µ) .
Then it is straightforward to estimate the matrix Γ t by substituting theoretical versions of the moments by their empirical counterparts. Let Γ t denote this estimator. One therefore obtains the estimated EDR direction b t as the eigenvector associated with the largest eigenvalue of Σ -1 Γ t where Σ is an estimator of Σ.

Population version of SIRdatastream. Let us denote by b t the EDR direction obtained in the block t. We consider the matrix

M T = T t=1 w t b t b t cos 2 (b t , b T ),
where the w t 's are positive weights such that T t=1 w t = 1. Under the assumptions of our model, the term cos 2 (b t , b T ) is equal to one since b t and b T are both colinear with β. Under the assumptions of our model, it is also straightforward to show that the principal eigenvector of M T is colinear with β and then is an EDR direction.

Let us remark that it is possible to reformulate this approach as the following optimization problem:

max v∈ p T t=1 w t cos 2 (b t , v) s.t. ||v|| = 1.
(

Indeed since ||b t || = 1, we have:

T t=1 w t cos 2 (b t , v) = T t=1 w t < b t , v > 2 = T t=1 w t v b t b t v = v T t=1 w t b t b t v = v M T v,
thus maximization problem (2) can be rewritten as max

v∈ p v M T v v v . (3)
The solution of ( 3) is clearly the normalized principal eigenvector of M T . Let us denote by v T this eigenvector.

Sample version of SIRdatastream. For t = 1, . . . , T , let us denote by bt the estimator of the EDR direction calculated on each block t. The estimator v T of the EDR direction v T with the SIRdatastream approach is the principal eigenvector of the p × p matrix defined as

M T = T t=1 w t bt b t cos 2 ( bt , bT ) (4)
where

w t = nt T j=1 nj and cos 2 ( bt , bT ) = ( b t bT ) 2 ( b t bt)×( b T bT ) .
Asymptotics. The following assumptions are necessary to state our asymptotic result for a fixed number T of blocks and a sample global size n which tends to ∞. Let n h,t be the number of observations in the hth slice in the block t and let n t = Ht h=1 n h,t be the number of observations in the block t. • (A1) Each block t is a sample of independent observations from the single index model (1). • (A2) For each block t, the support of Y is partitioned into a fixed number H t of slices such that p h = 0, h = 1, . . . , H t . • (A3) For t = 1, . . . , T and h = 1, . . . , H t , n h,t → ∞ (and therefore

n t → ∞) as n → ∞.
For each block t and under the assumptions (C), (A1)-(A3), from SIR theory of Li [START_REF] Li | Sliced inverse regression for dimension reduction (with discussion)[END_REF] each estimated EDR direction bt converges to b t at root n rate: that is, for t

= 1, . . . , T , bt = b t + O p (n -1/2 ). Since cos 2 ( bt , bT ) = cos 2 (b t , b T ) + O p (n -1/2 ) = 1 + O p (n -1/2 ), we get M T = M T + O p (n -1/2
). Therefore the principal eigenvector of M T converges to the corresponding one of M T at the same rate:

v T = v T + O p (n -1/2
). Since v T is colinear with β, then the estimated EDR direction v T converges to an EDR direction at root n rate.

Computational complexity. For sake of simplicity, let us assume that each block has the same sample size n * . One can show that the SIRdatastream approach performs faster than usual SIR based on the union of the first T blocks (called SIRglobal hereafter) provided that the sample size n * is large enough: n * > 2(p + 1). Moreover, when the total number T of blocks increases, the dataset used for SIR becomes larger and larger, and problem of data storage may appear. On the contrary, our SIRdatastream approach only needs the storage of the last block (necessary of the two last ones if a drift seems to occur) and the previous estimated EDR directions (which are only p-dimensional vectors).

Running time. We compare the running time (in seconds) of our SIRdatastream approach with SIRglobal (based on the union of the first T blocks).

For various values of the dimension p, the size n * of each block and the total number T of blocks, we generate B = 20 data streams and we evaluate the computational times for the two methods. Unsurprisingly one can observe in Figure 1 that the dimension p noticeably favours SIRdatastream versus usual SIR while the number T of blocks and the block size n * hugely penalize the usual SIR approach in comparison with SIRdatastream.

A simulation study

We consider for each block of data the same following semiparametric regression model:

Y = (X β) 3 + , (5)
Mean of computational times according to T when n * = 200 and p = 10 where X follows the p-dimensional normal distribution N p (0 p , Σ) with the covariance Σ arbitrarily chosen, follows the normal distribution N (0, 1) and is independent of X. For the slope parameter β, we consider various scenarios. For each scenario, we generate T = 60 blocks of size n * = 200 with p = 20.

• Scenario 1: β is constant for all the T blocks. We fix β = β 0 with β 0 = (1, -1, 2, -2, 0, . . . , 0) . • Scenario 2: β is constant for T -1 blocks and the 10th block is aberrant.

We fix β = β 0 for each block t with t = 10 and we set β = β 1 for the 10th block with β 1 = (1, 1, . . . , 1) .

• Scenario 3: β = β 0 for the first 9 blocks and β = β 1 for the remaining 51 ones.

• Scenario 4: β = β 0 for the first 9 blocks and β takes different values for the remaining 51 blocks. The 51 slope parameters β have been randomly generated as follows: each component β j of β is randomly obtained from the normal distribution N (0, 1).

We use the following quality measure for any estimator (denoted by β) of the direction β:

cos 2 ( β, β) = ( β β) 2 ( β β)×(β β)
. The closer to one is this measure, the better is the estimate.

For each scenario, we generate T = 60 blocks as described above. Then at each time t (i.e. when the first t blocks are available) we estimate the EDR direction with SIRdatastream and SIRglobal approaches. We also estimate the EDR direction with usual SIR based only on the data of this block t.

In the following, for each scenario, we plot the quality measure cos 2 ( β, β 0 ) (resp. cos 2 ( β, β 1 )) on Figure 2 (resp. 3) of the estimator β (obtained with SIRdatastream, SIRglobal or SIR estimators at each time t). We also represent with an image the weights cos 2 ( b t , b T ) used in equation ( 4). The lighter is the color, the larger is the weight. This image will provide to the user an interesting graphic in order to detect if a drift occurs or if aberrant blocks appear in the data stream. 2 and3.

Some comments on Figures

• SIRdatastream and SIRglobal perform well for scenario 1 (but keep in mind that SIRdatastream is an efficient method from running time and data storage points of view). The image of the weights does not exhibit any drift or aberrant block.

• For scenario 2, SIRdatastream and SIRglobal perform well except for the 10th block for SIRdatastream. But the image of the weights clearly indicates that this block is aberrant and then the effect of this block on the SIRdatastream estimator disappears when the new blocks are available.

• For scenario 3, the image of the weights clearly shows that there is a drift from the 10th block to the last one. The estimation of the true direction β 0 is efficient for SIRdatastream and SIRglobal for the first 9 blocks and then becomes worse for the next blocks. Note that in Figure 3 one can see that SIRdatastream is efficient to estimate the true direction β 1 from the 10th block to the last one, this is not the case for SIRglobal.

• For scenario 4, the image of the weights clearly indicates that there is no common structure from the 10th block to the last one. The estimation of the true direction β 0 is efficient for SIRdatastream and SIRglobal for the first 9 blocks and then becomes worse for the next blocks. One can remark that SIRglobal always provides estimates close to the direction of β 0 after the 10th block even if there is no structure in this case, which may cause troubles in practical situations. This remark also remains valid for scenario 3 where SIRglobal provides estimates close to the direction of β 0 after the 10th block even if the new underlying direction is β 1 . At the end of the 60th block in scenario 3, SIRglobal is still not able to provide a good estimate for β 1 , see Figure 3.

Concluding remarks

The proposed approach performs well on simulated data. We present in this paper a single index version of the underlying model. It is possible to extend this approach to multiple indices model. In this expression of M T , the vector b t (resp. the squared cosine) will be replaced by a basis B t of the EDR space (resp. a proximity measure between two K-dimensional EDR space, for instance the square trace correlation). It is also possible to use alternative methods instead of SIR (such as SIR-II, SAVE or SIR α for example). In the next future, SIRdatastream will be applied on real data dealing with the estimation of Mars surface physical properties from hyperspectral images.

4 TFig. 1 .

 41 Fig. 1. Running times (in seconds) for various values of p, n * and T .

Fig. 3 .

 3 Fig. 3. Scenario 3: plot of the quality measure cos 2 ( β, β1) versus the number T of blocks (red for SIRdatastream, green for SIRglobal)

  Scenario 1: a common direction in all the T = 60 blocks 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60Fig.2. Numerical behaviour of the SIRglobal and SIRdatastream estimators for various scenarios. On the left: plot of the quality measure cos 2 ( β, β0) versus the number T of blocks (red for SIRdatastream, green for SIRglobal on the first t blocks, black for SIR on block t only). On the right: image of the weights cos 2 ( bt, bT ) used in the computation of vT .

		1.00 1.00 1.00							9 5 1
	quality measure quality measure quality measure	0.75 0.80 0.85 0.90 0.95 0.75 0.80 0.85 0.90 0.95 0.75 0.80 0.85 0.90 0.95							59 54 49 44 39 34 29 24 19 14
		0 0 0	10 10 10	20 20 20	30 number of blocs 30 number of blocs 30 number of blocs 2 4 6 8 Scenario 2: the 10th block is aberrant 40 50 60 40 50 60 40 50 60
		1.0 1.0 1.0							9 5 1
	quality measure quality measure quality measure	0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8							59 54 49 44 39 34 29 24 19 14
		0 0 0	10 10 10	20 20 20 2 4 6 8 Scenario 3: a drift occurs from the 10th block (β0 to β1) 30 40 50 60 number of blocs 30 40 50 60 number of blocs 30 40 50 60 number of blocs
		1.0 1.0 1.0							8 6 4 2
	quality measure quality measure quality measure	0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8						Nombres de blocs precedents	59 56 53 50 47 44 41 38 35 32 29 26 23 20 17 14 11
		0 0 0	10 10 10	20 20 20	30 30 30	40 40 40	50 50 50	60 60 60	2 3 4 5 6 7 8 9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57	59
					number of blocks number of blocks number of blocks														Numero bloc courant
		Scenario 4: from the 10th block to the last one, there is no common direction β
		1.0 1.0 1.0							9 5 1
	quality measure quality measure quality measure	0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8							59 54 49 44 39 34 29 24 19 14
		0 0 0	10 10 10	20 20 20	30 30 30	40 40 40	50 50 50	60 60 60	2 4 6 8
					number of blocs number of blocs number of blocs


D Hestroffer
email: hestroffer@imcce.fr

Science from NEOs -Limitations and Perspectives

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The Gaia mission, in addition of doing astrometry of stars and a 3D census of our Milky Way, has specific objectives for solar system objects [START_REF] Tanga | Gaia FUN-SSO workshop[END_REF]. In that respect it follows the Hipparcos/Tycho mission that already observed a few solar system objects. The astrometric accuracy involved with Gaia is however 1 to 2 orders of magnitude higher, and the total number of targets observed is incomparably larger. Here we describe several scientific outcomes from the astrometry of asteroids and comets with particular emphasis on the Near Earth Objects (NEOs), their specificities and limitations.

Astrometry from Space -Hipparcos & Gaia

The ESA Hipparcos/Tycho mission (1989-1993:1997) observed 48 asteroids, a few planetary satellites, and two major planets. It basically provided positions at 10mas precision level and photometry at 0.05 mag level, which data are catalogued in the Hipparcos solar system annex, and available at the CDS1 [START_REF] Esa | The Hipparcos and Tycho Catalogues[END_REF]. These observations provided some scientific output (see [START_REF] Perryman | Astronomical Applications of Astrometry: A Review Based on Ten Years of Exploitation of the Hipparcos Satellite Data[END_REF], and reference therein), we will emphasise here that already at that level of astrometric precision the photocentre effect is visible [START_REF] Hestroffer | Photocentre displacement of minor planets: analysis of HIPPARCOS astrometry[END_REF], and that photometric inversion with sparse data can be applied to targets with enough observation points [START_REF] Cellino | Genetic inversion of sparse disk-integrated photometric data of asteroids: application to Hipparcos data[END_REF]. Besides, the availability of the Tycho catalogue of stars has also been valuable for the science of solar system objects by improving astrometric reductions and stellar occultations predictions. In contrast, the Gaia mission is much more than just a Hipparcos II.

One major technical difference with Gaia comes from the use of CCDs is TDI mode (Hipparcos used an old technology of photomultiplier and a modulating grid) which enables to simultaneously observe all objects brighter than magnitude V≤20 in the field of view (given it is not in a over-crowded region in the galactic plane). Gaia will also provide photometry in a large G-band as well as low resolution spectrophotometry (high resolution spectroscopy is obtain over a very small range and of little use for asteroids except for the purpose of calibrating the instrument). There will be about 250.000 asteroids observed, about 25 planetary satellites, and JFC as well as LP comets. The images on the CCD will enable to derive sizes of a few thousands of asteroids to compare to those of the IRAS catalogue, and shape or binarity information for several. The photometry will enable to derive the spin state (direction and period) as well shape information (tri-axial model) for almost all observed body. Of course, being an astrometric mission, the main output will be the astrometry. Gaia will provide high accuracy positions for all asteroids over 5 years and directly connected to a homogeneous reference frame of distant quasar (and hence connected to the ICRF).

Gaia

As said before the limiting magnitude is V≤20 (besides the fact that very bright objects won't be observed, as well as in very crowded region a selection will be applied, but all this does not affect asteroids much). There is also a limit of size at about 0.7-0.9 arcsec; objects larger than that will not be detected and hence not be observed, this concerns mainly planetary satellites. Fast moving object should be detected but they might not be observed through all the 9 CCDs of the FOV, unless a specific windowing scheme is applied. The given satellite's scanning law makes that the objects are observed when they cross the FOV (this is no pointing telescope) which results in a random pattern of observation distribution. Nevertheless the observations are rather well distributed in time and in space for a typical main belt asteroid, with 60-70 observations on the average over 5 years (Fig. 1). The situation is not the same for a NEO or an object close to the limiting magnitude, where observations can be scarce and clustered over a short period of time. Binary asteroids will also be observed by Gaia enabling improvement of their mass, or determination de novo. The cases of binaries can be of particular interest for Gaia, whether the two components are well separated and detected as two different bodies (resolved binary) or whether the system appears as a single object but the motion of the photocentre shows a wobble effect around the centre of mass that is not correlated with the spin of the body (astrometric binary). Further work should be carried out to determine how many resolved or astrometric binaries will be detected as such. The detection of an astrometric binary necessitates to have good absolute positions, with good precision and hence a good stellar catalogue, which Gaia will provide. The signature depends also on the separation of the bodies and mass ratio; it is most sensitive for mass ratio of the order of 0.05-0.5. Application to a known transneptunian binary (TNB) system have shown some interesting potential (Ortiz 2010). Gaia will also observe a handful of trans-Neptunian objects among, which the Pluto/Charon system. Making use of the GIBIS focal plane simulator [START_REF] Babusiaux | The Gaia Instrument and Basic Image Simulator (GIBIS). In : The Three-Dimensional Universe with Gaia[END_REF]) and the Gaia scanning law rendez-vous simulator, it is found that about 70 relative positions will be acquired of the system as a resolved binary, which-when combined to future or past astrometric observations-will enable to monitor the orbit evolution over several decades. Making use again of the GIBIS simulator, it is found from a population of synthetic binaries, that system with a separation larger than 0"3 and a magnitude difference smaller than 2 (going up to ∆mag=5 at 1" separation, see Fig. 2) will be resolved by the sky mapper. There are several timelines involved in the Gaia mission either in the data reduction pipeline [START_REF] Berthier | Gaia FUN-SSO workshop[END_REF] or in the data release (intermediate or in alert, [START_REF] Pristi | Gaia FUN-SSO workshop[END_REF]. For what concerns the long-term treatment of the astrometry of solar system objects, there will be basically two kind of outputs: The catalogue of measured positions (asteroids, comets, and satellites) in the final reference frame, and also derived parameters concerning either a specific object (local parameters) or the solar system and physical constant in general (global parameters). We describe in the following some of the aspects of the parameters estimation.

Orbits

Gaia will provide astrometry of solar system objects on a transit level basis, on the order of 0.3 mas for objects brighter than V ≤14 mag, going up to approx. 3 -5mas at the magnitude limit V=20. Because a fast moving object will not be observed through its whole FOV transit, we consider in the following the pessimistic case where only one CCD observation is available per crossing. On the other hand, the ephemerides are computed by numerical integration of the perturbed two-body problem, taking into account the perturbations of the planets. The computation additionally takes into account previously selected perturbing asteroids, relativistic effect (for the equation of motion and observation reduction), and the photocentre offset due to phase. In case of comets, we also include non-gravitational forces through empirical A i parameters. Last, we also perform the integration of the variational equations to derive the partial derivatives with respect of our parameters to estimate. Once the O-C vector and Jacobian matrix are computed, we perform a linear least-squares inversion.

The astrometry acquired will enable orbit improvement by a factor ≈30 when compared to what is obtained presently from ground-based data only. In the case where the number of Gaia observations for a given object is insufficient however, there will be no particular orbit improvement but the data will be useful to derive global parameters, and later, for orbit improvement when combined to all existing ground-based observations. Besides given the relatively short period of observations, one will cover a full orbital period of a typical mainbelt asteroid, but not enough to derive longer-term effects as can be present for instance in the planetary satellites motion, or Trojans long term librations.

Mass of asteroids can be derived from the monitoring of the relative orbit in binary system as seen before, but also from monitoring the orbit of a target asteroid when perturbed by a massive perturber asteroid; there will be about 150 mass determination of asteroid involving several thousands of asteroid-asteroid close encounter. In some cases the mass determination of a few asteroids can be enhanced by complementary ground-based astrometric observations [START_REF] Ivantsov | Gaia FUN-SSO workshop[END_REF], the same holds for the density determination by considering additional observations from ground to better characterise the shape and size of the object. Other global parameters entering the dynamical model for the equations of motion can also be adjusted.

For instance by monitoring the precession rate of all the asteroids' perihelion together, one will enable to derive the relativistic PPN parameter ß as well as a direct estimation of the solar dynamical flattening J 2 . Besides one will be able to derive the link of the dynamical reference frame to the Gaia (and hence ICRF) reference frame, and a possible time variation dG/dt of the gravitational constant (Hestroffer et al. 2009).

NEOs

The astrometry of asteroids will also enable orbit determination or linking in case a new object is discovered. Given the modest limiting magnitude when compared to ongoing and future ground-based surveys, Gaia will not discover a huge amount of asteroids. However being in space and free of atmospheric diffusion and airmass, it might be more efficient at discovering objects at low solar elongations (down to 45°) and in particular objects orbiting inside the orbit of the Earth (IEOs). Orbit determination in this case will be done on the shortterm level by use of an MCMC algorithm which will provide a whole set of possible solutions [START_REF] Granvik | OpenOrb: Open-source asteroid orbit computation software including statistical ranging[END_REF]. In such cases it is of importance to have additional observations and astrometry in support to avoid loosing the object and help further identification when the object crosses back the satellite's FOV several weeks or months later (Thuillot 2010).

As seen in [START_REF] Bancelin | Gaia FUN-SSO workshop[END_REF] there will be about 1600 NEOs observed; only objects that can be identified will enter the long-term reduction pipeline. The number of observations per object is varying (about half of them will have less than 10 observations), but even if no orbit improvement can be obtained, they all enter into the global effects parameters estimation scheme. Moreover the NEOs with low semi-major axis and large eccentricity are the most sensitive to the General Relativity test. In that case, combination of long-term observationsbecause the effect is a secular drift -gives a higher leverage for the test. We shall here consider combination of present radar [START_REF] Margot | [END_REF] with future Gaia observations. For small objects close to the Sun the Yarkovsky effect and other non-gravitational effects can be noticeable. In a few cases a simple inversion can be performed to provide a basic scaling parameter for the Yarkovsky effect to better than 20%. This can be the diameter and thermal inertia, all other parameters of the model being assumed; or more generally a parametric force in the along track direction that is inversely proportional to the square of the heliocentric distance. Here again ground-based observation can be useful to constrain the dynamical system to be used.

Conclusion

Gaia will observe a large number of solar system objects, much more than Hipparcos and Tycho could do. These will essentially be main belt asteroids, but about 1600 Near Earth Objects will be observed too, including some inner-Earth objects that might be discovered by Gaia. The astrometry, photometry, and colour-spectroscopy acquired will enable direct output by considering the Gaia data alone (Fig. 3). In some cases complementary ground-based data is mostly valuable.

Fig. 1 -

 1 Fig. 1 -Observation distribution in time and space for typical asteroids. The total number of observations can decrease to a few for NEOs and objects close to the limiting magnitude.

Fig. 2 -

 2 Fig. 2 -Binaries detection limit in ASM. Simulation form a synthetic population as a function of separation and magnitude difference (credit J. Blanchot).

Fig. 3 -

 3 Fig. 3 -General scientific output from the Gaia observation of asteroids (credit: P. Tanga).

URL: http://cdsarc.u-strasbg.fr/viz-bin/Cat?I/239

Acknowledgement

The author wishes to thank D. Bancelin, J. Blanchot, P. David (IMCCE) and the colleagues of the DPAC CU4 SSO group for their help, discussions, and inputs.


S Bouquillon

F Taris

M Altmann

Gal Jose Cristino

SYRTE and PARSEC Contribution for the GBOT/GAIA Moving Target Astrometry

Introduction

GAIA will measure to unprecedent precision positions, movements, and parallaxes, by the superposition of two fields apart by 174deg, taken from the L2 Earth-Sun, about 1.5 million km from the ground. To achieve the aimed precision for stars, and particularly for solar system bodies, the instantaneous position and speed of the satellite must be known respectively to 150m and 2.5 mm/s. This translates to the GBOT (Ground Base Optical Tracking) requirement to deliver quasi-daily positions of the satellite at the accuracy of 10mas relatively to the GAIA´s reference frame itself (Altmann et al., 2010, this proceeding). The challenge increases because the satellite will probably be dimmer than R 17th magnitude and will be moving on average at 30mas/s, and switching hemispheres between summer and winter. We will present the strategies worked out for the satellite centroid´s determination, including tracking mode, binning, super-gaussian fit, blind co-addition of images; as well as the astrometric reduction open code designed to cope with this variety of conditions. We will show applications of these resources to observations of the satellites WMAP and PLANCK, and to fast asteroids. All these topics, on the other hand, make for an exchange of experiences and ensuing of scientific programs with other groups targeting fast, dim objects prior and during the GAIA mission.

Goals and feasibility

The main issues of the campaigns aiming to emulate the observation conditions of the GAIA satellite are speed, magnitude, and their variations. For the observational band there were two main a priori choices, between V-band or clear against the near infrared. In the first case maximum light is gathered, but atmospheric extinction and chromatic refraction are also large. On the near infrared atmospheric extinction is less important but the satellite and the reference stars are dimmer, demanding longer exposures. A compromise on the R-band, which is readily available from all envisaged observatories, favors also the closeness to GAIA's band. In particular the early studies made with WMAP have shown the important magnitude modulation brought by the satellite´s spin.

Here, we focus on the other important issue, which concerns the astrometric pipeline, given the precision required and movement of the target. All this considered, observations are being taken of satellites within the Lagrange zone L2 (WMAP and PLANCK), and asteroids with large tangential motion (mostly double or suspected, in order to mimic the satellite spin).

Astrometric handling

The astrometric pipeline is preceded by a database, which rectifies headers´ inconsistencies and makes the images available for reduction. Specific routines are being developed for the GAIA´s case. The first step is the recognizing of objects, using variable thresholds and local boundaries of search and adjustment. The routine can deal with crowded fields as well as with fields affected by Moon´s illumination gradient. In particular, it is able to recognize GAIA either as an elongated or particularly dim object. The derivation with the equatorial coordinates can use pre-existing catalogs of earlier version of the GAIA catalog itself.

Blind co-addition

The Moving Gaussian approach (Fig. 3) describes to mathematical exactness the displacement of the GAIA satellite or, for instance, of an asteroid. To tackle the problem of the intrinsic faintness of the object, another approach is proposed, which we term as Blind Co-addition. It borrows from the standard co-addition procedures used to pile up images, enhancing the signal to noise ratio of faint objects without worsening the centroid´s determination. The distinctive feature of the blind co-addition when applied to enhance a moving target is that it cannot no longer rely on the reference of the bright stars positions, since the target position is varying with respect to them. Thus, the piling up of the target images relies on the constancy of its underlying distribution of incoming photons. In practice, though the PSF shape is disrupted, the barycenter of the illuminated region around the object would remain the same. The blind co-addition procedure entails taking short images, so that a large number of references stars are still well sampled and for which the centroid can be determined to derive the required astrometric precision, orientation and scale. At the same time the moving target remains effectively circular. The first step of the blind co-addition procedure is to take several frames, preferably in a fast sequence, up to theoretically being able to reach the signal to noise ratio goal. Next a region around the moving object is cropped centered on the ephemeris position. For each frame the orientation and pixel scale are obtained by the astrometry of the well imaged stars. The ensemble of orientation and pixel scales values (in practice much alike) are used to normalize all of the target´s regions coming from the several frames. If the target motion is not linear, the regions orientation is also compensated for. Then in each region the barycenter of the illuminated portion is calculated. All regions are now co-added by the barycenter, at the nominal precision aimed at, and a bi-dimensional Gaussian is adjusted to the resulting distribution. The centroid is de-convoluted towards the orientation and scale of each frame, so that the original region is replaced by the compound one, but particular to the original frame instead of common to all of them. In this way one independent measurement is extracted for each frame. The independent measurements can finally be averaged, either using the ephemeris speed if it can be taken as correct for the total time range, or leaving a correction to the ephemeris speed as unknown.

Results and Conclusion

A series of observations of PLANCK was conducted at the 2m. Liverpool Robotic Telescope, at La Palma, from 10 th to 18 th August 2010. The integration time was 20sec and a sequence of 10 measurements was made daily. On these conditions, namely, brighter probe, longer integration time, and limited number of daily observations, the blind co-addition still improved the correction to the ephemeris. Relatively to a local frame of 2MASS stars, on the best observation night the error on R.A. went to 46mas and the DEC. error went to 58mas, respectively a reduction of 8mas and 2mas in comparison with the average of individual reductions. On the worst night, however, the same comparison shows an improvement on R.A. by 35mas but a worsening on DEC. by 24mas.

Fig. 5 -Compound images of PLANCK taken at the 2m LT at La Palma, for series of 10 exposures of 20sec. On the left the night of best seeing is depicted, and on the right that one of worst seeing. Though nominally the error on the centroid is nearly the same (1mas), on the worst night there was no net improvement upon the results from the average of standard astrometric reductions.

Both the Moving Gaussian and the Blind Co-addition offer a substantial improvement for the determination of the centroid of the GAIA satellite during its mission. Their utilization in the astrometric pipeline used by the GBOT/GAIA will hence depend of the actual satellite brightness, and variation, during the mission, as well of observational conditions, like seeing, number and length of exposures. The series of tests using PLANCK and selected asteroids will continue to improve the present procedures.

Fig. 1 -

 1 Fig. 1 -On the left the very first images of the test campaigns, taken of the WMAP satellite on April 5 th , 2008. On the right light curve of double asteroid Barbara234, taken on December 19 th , 2009, and which is part of a larger campaign for this object. Both observations were made at the ESO2.2m/WFI telescope, in Chile.

Fig. 2 -

 2 Fig. 2 -On the left an example of object recognition in a dense field and wildly different magnitudes. On the right the improving on the centroid determination of a moving object from a barycentric fit (outer circle) to a fitting through a bi-dimensional Gaussian plus a displacement term (inner circle).

Fig. 3 -

 3 Fig. 3 -For the simulated moving object appearing on the left, the middle figure shows the residuals for a fitting by a bi-dimensional Gaussian, while the left figure shows the much smaller residuals resulting from the fitting by the Moving Gaussian procedure developed for the GAIA (and asteroids) case.

Fig. 4 -

 4 Fig. 4 -Principle of the Blind Co-addition. From an actual observation of WMAP, with 12sec exposition, at the ESO 2.2m/WFI, on the top panel it is seen the local region (left) and reconstructed photon distribution (right). On the lower panel are the corresponding compound region and photon distribution of the co-addition of 20 frames.


J.-E Arlot

The Evolution of the Networks of Observers of Phenomena

Introduction

Astrometric observations are commonly measurements of angular positions of bodies on the celestial sphere in a given reference frame. However, the precision of such measurements is limited by the instruments and by the receptors used for these observations. Astronomers saw very early that celestial bodies were often involved in phenomena such as eclipses of the Sun or the Moon. Such phenomena correspond to specific positions of the involved bodies in space. At least, one may say that the topocentric positions of the Moon and the Sun are very close at the time of an eclipse. This is an astrometric observation. Since the possible phenomena are numerous in the solar system thanks to the velocity of the moving objects, astronomers made predictions of such events and made observations as precise as possible in order to deduce astrometric positions from these events. Of course, a model was necessary for that purpose but this was the main way to improve our knowledge of the solar system at the beginning of astronomical research.

What phenomena?

Jovian satellites eclipses

After the eclipses of the Sun and the Moon, the phenomena the most observed by the astronomers were the eclipses of the satellites of Jupiter by the planet Jupiter. As soon as Galileo observed these satellites, he understood the phenomenon of the eclipse occurring very often (Io makes a revolution around Jupiter in one day and an half and is eclipsed each time!). It was then easier to date each eclipse in order to get the period of the satellites than measuring the relative positions of the satellites.

Fig. 1 -The principle of the phenomena of the Galilean satellites of Jupiter

From Galileo until the beginning of the XX th century, Jovian eclipses of the satellites were extensively observed and were the basis for the building of the dynamical models of the motion of the Galilean satellites. The theories by Laplace (end of the XVIII th century) and Sampson (beginning of the XX th century) are among the most achieved analytical models and are based on eclipses. However, the precision of these observations is limited by the refraction of light in the upper atmosphere of Jupiter determining the shadow cone. The progress of the direct astrometry thanks to the apparition of the photographic technique led to the decline of the use of eclipses. During the XX th century, the photographic technique was the main source of observations of the Galilean satellites.

Mutual events of the natural planetary satellites

As seen on figure 1, the eclipses by Jupiter are not the only phenomena occurring in the Jovian system: the satellites may eclipse and occult themselves since they are in the same orbital plane. Then, when the Sun (and the Earth which is close to the Sun as seen from Jupiter) crosses the common orbital plane of the satellites, mutual eclipses and occultations occur. Note that this occurs at the equinox on Jupiter since the orbital plane of the satellites is the equatorial plane of Jupiter. The interest of these events is that the satellites have no atmosphere so that the precision of the observations is higher than the one of the eclipses by Jupiter and also higher than the one of the photographic data. However, these observations started only after 1973 because of the need of computers to make the prediction of the events. These events occur not only in the Jovian system but also in the systems of Saturn and Uranus since they have large satellites orbiting in their equatorial plane. Note that these events occur at the equinox on the planet i.e. every 6 years for Jupiter, 15 years for Saturn and 42 years for Uranus. These observations are then a complement of high accuracy to the direct ground based astrometric measurements as data from space probes available only on short intervals of time. It is easy to understand that these events occur at any time and that the observations are possible only from selected geographic area. More, the observers may not wait for favorable weather, so that an organized network of observers is necessary.

Occultations of stars

All the moving objects of the solar system may occult a star on their way on the celestial sphere. Such an occultation is similar to a solar eclipse, the Sun being replaced by a star and the Moon by any moving object. For small objects for which the resolution of the telescopes does not allow to measure the size, each occultation provides one measurement of the size of the object. Several occultations observed from close sites will provide a complete profile of the object. As for solar eclipses, the total occultation is observable only from a central path where the observers should be placed. A network of observers is necessary to get useful observations.

Mutual events: the history

Galileo observed the first eclipse by Jupiter in 1612 but it was only in 1693 that Arnoldt observed an occultation of Europa by Ganymede. Such observations occurred only by chance when observing the satellites but the calculation of prediction was not possible at that time. The calculations were difficult because of the sensitivity of these events to the accuracy of the position of the satellites near 100 km. Such accuracy needs the use of a complete dynamical model for the calculations. At the beginning of the XX th century, the Sampson's theory was sufficient for such predictions but the algorithm needed too many calculations. From the 1970's, computers were used for astronomical calculations and precise predictions of mutual events were published [START_REF] Arlot | Les occultations et les eclipses mutuelles des satellites galiléens de Jupiter[END_REF], Brinkman, 1973). Since these events were rare and could provide relative positions and diameters of the satellites, observers were numerous to try to catch some events during the favorable occurrence of 1973. Even the Galilean satellites are bright, the recording of the events needed a fast photometric receptor associated to a telescope the aperture of which being larger than 50 cm: an occultation or an eclipse was only a few minutes long. Since made by photometrists, the observations were of good quality, well calibrated. The only problem was to be sure of the time scale: each event must be observed in the Universal Time scale in order to be linked to the other events and to the theoretical model.

The observers, the material and the network

The first observers

The first observers were professional astronomers using 50cm-telescope or larger. They were photometrists using a photoelectric photometer. Some amateurs tried to make visual observations using the methods developed for variable stars observation. Table 1 provides the evolution of the telescopes and receptors used for the observation of the mutual events. Seven Jovian occurrences allowed the observation of the mutual events of the Galilean satellites and about 1800 observations were made. At the beginning, large telescopes managed by professional astronomers equipped with single channel photoelectric photometers were the more numerous systems of observations. From 1985, 2D receptors such as CCD cameras appeared and were used allowing recording a reference object at the same time than the occulted or eclipsed satellites: observations were possible even in difficult conditions such as twilight or fog. The problem was to record images with a high frequency (more than one image per second) that was difficult at the beginning of the use of the CCD's. The progress of that type of 2D receptors led to the disappearance of the 1D receptor for the 2009 occurrence.

Correlatively, the part of amateur's observations grew rapidly due to increase of the sensitivity of the receptors allowing using small telescopes. Specific training of the observers was made in order to learn the basis of photometry and also to understand the need of the use of an accurate time scale linked to UTC.

The observers today

To day, small fast CCD cameras such as Watec are widely used associated to 20 or 30cmaperture telescopes: such material is easy to get and the number of observing sites of the network increases. Nowadays, the network (cf. figure 2) allows observing as many events as possible and includes ninety percent amateur astronomers. Internet provides help and software for the reduction to the observers and images are broadcasted through the Web.

Fig. 2 -The present PHEMU network of observers

The Saturnian and Uranian events

The success of the observation of the mutual events of the Galilean satellites led to try the observation of the same events for the Saturnian and Uranian satellites. For those systems, some difficulties arose: the field was smaller because of the increased distance of the satellites from the Earth leading to a smaller apparent distance satellites-planet and the bright planet (plus ring for Saturn) made difficult the observations. More, for the Uranian satellites, their faint magnitude made the use of large telescope necessary. For example, only one event was observable through a small telescope in the visible wavelength thanks to its distance to the planet at the time of the event: all the other events occurred too close to the planet Uranus, too bright in the visible wavelength. Then, these events needed the use of infra red filters making the planet very dark. However, such K' filter implied the use of large telescopes from the 3.5m-NTT (cf. an image in figure 3) to the 8m-VLT (Arlot et al., XXX). The first use of the observations of mutual events has two purposes: the determination of relative positions between two satellites and the measure of the radii of the satellites (at that time, the space probes had not yet provide these data). These quantities were correlated but numerous observations permitted to de correlate them. After the accurate determination of the radii by the space probes, it appeared that the mutual events were sensitive to the law of reflection of the light at the surface of the satellites which was the explanation of non symmetrical light curves. More, it appeared that the observation in infra red should show the hot spots at the surface of Io (Descamps et al., XXX) through their occultation (figure 4). Nowadays, the hot spots may be observed through specific 2D infra red receptors. The analysis of the light curves allows now to determine highly accurate relative positions of the satellites used for the fit of the new theoretical dynamical models. After 30 years of observations of mutual events and one century of photographic observations, small effects in the orbital motion of the satellites such as tidal effects may be detected (Lainey et al., XXXX, Lainey et al. 2009).

Reusing the old data

Table 2 provides the number of observations gathered since the beginning of the observational campaign. The first light curves were analogical and their analysis very simple. It is possible to digitize them and to apply a better algorithm to determine the relative positions of the satellites. The search for older data allowed us to find in publications the measurement of the time of conjunction between satellites with a poor accuracy, not useful nowadays.

Conclusion

The example of the network of observers of the mutual events of the natural planetary satellites shows an evolution which depends on the progress of the receptors. More and more amateur astronomers may participate to the observations of phenomena that allows the increase of the number of observations. The network of observers of occultation of stars, using the same material, are more dependant of each event, observable only on a small area: in this case, the observers must move towards the best observing places. For any other observational campaign, specific constraints may appear but the technical progress of the material to be used and of the communication through Internet makes the things easier for observers.

Fig. 3 -

 3 Fig. 3 -The Uranian satellites as seen with the NTT in the K'-band. The planet Uranus is darker than the satellites (except the small Miranda)

Fig. 4 -

 4 Fig. 4 -Occultation of Io by Europa: the occultation of the hot spots are visible on the light curve at right

Table 1 -

 1 Evolution of the size of the telescopes and of the receptors

	Occurrences	Size of the telescopes	Photometry
		< 60cm	> or = 60cm	1 D	2 D
		(amateurs) (professionals)
	Jupiter
	1973	4	20	24	0
	1979	3	7	10	0
	1985	12	12	21	3
	1991	37	19	39	17
	1997	35	10	15	30
	2003	34	15	8	41
	2009	52	10	0	62
	Saturn
	1980	0			0
	1995	5	11	8	8
	2009			0
	Uranus
	2007	4	11	0	15

Table 2 -

 2 Observations made since the beginning of the campaigns

		Number of	Number of	Number of	Number of
		observations	observing sites	observed events	observable events
	Jupiter
	1973	91	26	65	176
	1979	18	7	9	60
	1985	166	28	64	248
	1991	374	56	111	221
	1997	275	42	148	390
	2003	361	42	116	360
	2009	523	68	206	237
	Saturn
	1980	14	6	13	213
	1995	66	16	43	182
	2009	26	15	17	131
	Uranus
	2007	52	19	36	193


Hestroffer, D D Bancelin

Near-Earth Asteroids Astrometry with Gaia and Beyond

Near-Earth Asteroids Astrometry with Gaia and Beyond

David Bancelin, Daniel Hestroffer, W.

Introduction

Gaia is an astrometric mission that will be launched in 2012 and will observe a large number of Solar System Objects down to magnitude 20. The Solar System Science goal is to map thousand of Main Belt Asteroids (MBAs), Near Earth Objects (NEOs) (including comets) and also planetary satellites with the principal purpose of orbital determination (better than 5 mas astrometric precision), determination of asteroid mass, spin properties and taxonomy. Besides, Gaia will be able to discover a few objects, in particular NEOs in the region down the solar elongation (45°) which are harder to detect with current ground-based surveys.

In the first section, we detailled the nominal scanning law of Gaia and its impact on the number of observations of NEAs. Then we focus our study on asteroid Apophis were we analyze the effect of Gaia observations on the actual position uncertainty, and on the 2029target b-plane. In the last section, dedicated to the astrometry of newly discovered objects by Gaia, we analyze the combination of ground-based and space-based data on the short-term ephemerides.

Nominal Scanning Law of Satellite Gaia

During the 5-years mission, Gaia will continuously scan the sky with a specific strategy (fig. 1): objects will be observed from two lines of sight separated with a constant basic angle. Five constants already fixed determinate the nominal scanning law but two others are still free parameters: the initial spin phase and the initial precession angle. These latter will be fixed at the start of the nominal scientific outcome (possibility of performing test of fundamental physics) together with operational requirements (downlink to Earth windows). Several sets of observations of NEOs will hence be provided according to the initial precession angle. We used a Java rendez-vous simulator which provided us 35 sets of Gaia observations. Figure 2 shows the number of NEAs and PHAs that could be observed by Gaia. The number of asteroids does not really change according to the value of the initial precession angle. The mean values of possible observed asteroid are ~1650 NEAs and ~405 PHAs.

Study case of asteroid 99942 Apophis (previously 2004MN4)

We study here the effect of Gaia observations on Apophis orbit. This asteroid has a so deep close-approach with the Earth on April 2029 that its post close-approach orbit becomes chaotic. Thus, the uncertainty on the geocentric position and distance becomes large. From a linear propagation of the covariance matrix, figure 3 shows the impact of Gaia astrometry done in 2014 (date of last Gaia observations) on the position uncertainty of Apophis. To this purpose, we considered only one of the sets provided by Gaia. The position uncertainty is reduced to less than 2 km with Gaia observations. This value remains almost constant until the 2029 close approach (at distance 38000 km to Earth) where the uncertainty will start increasing. We can also analyze the impact of Gaia observations on the geocentric coordinates (ξ, ζ) of the 2029-target b-plane [START_REF] Valsecchi | Resonant Returns to Close Approaches: Analytical Theory[END_REF]. The b-plane passes through Earth's center and is perpendicular to the geocentric velocity of the asteroid. The initial covariance of the (ξ, ζ) elements are propagated to this date. Figure 4 represents the 3σ scattering ellipse where the semi minor axis is defined by 3σ ξ , the semi major axis by 3σ ζ and its center by the values of (ξ, ζ) on the nominal solution.

Fig. 4 -Scattering ellipse on the target plane on date of close approach (2029/01/13.907) with (blue) and without (grey) Gaia observations.

The uncertainty ellipse size is strongly reduced and the geocentric position of Apophis, at the date of closest approach, is better determinated, considering Gaia observations.

Astrometry for newly discovered objects

By combining, in real-time, ground-based to space-based data, it is possible to drastically improve the short-term ephemerides. Figure 5 shows an illustration of this improvement for the prediction of a newly discovered object by combining the two kinds of data. For our simulation, we considered a hypothetic Apophis that would be discovered by Gaia. When observing a new object, the satellite will send to Earth, as an alert, the coordinates of the unknown object. Thus, it is possible to make a prediction of the position of the hypothetic Apophis on the sky plane by computing a preliminary orbit (using Statistical Ranging method [START_REF] Virtanen | Statistical Ranging of Asteroid Orbits[END_REF]). This prediction was made three days after its discovery by Gaia and the 1σ distribution is large (1 degree) and quite far from the expected value (triangle). If we make a geocentric observation on the 4th day after its discovery and combine it with the late Gaia observations, the (α, δ) uncertainty is reduced by a factor 30 and the ephemeris is well improved (note that here the 10σ distribution is given). We can wonder now how many alerts are expected. To answer this question, we considered a sample of 20,000 of NEOs which could be either known or unknown objects. Figure 6 shows the statistic of the possible observed objects. This number appears to be around 12%. But if we compare this number to the one of known asteroid that will be observed by Gaia (~29%) we can just conclude that we have more chance to observe known objects than discovering new objects.

Fig. 6 -Statistic of objects that would be observed by Gaia among a 20,000 synthetic population.

Conclusion

Even if Gaia will not be a big NEAs discoverer, it will provide unprecedented accuracy for NEAs orbit's improvement. Besides, this study can be continued considering the astrometric reduction due to the stellar catalogue provided by Gaia. As a matter of fact, this catalogue will be more precise and dense and almost free of zonal errors. Thus, classical ground-based astrometry (and concerning hence more objects down to fainter magnitude) will be improved.

Fig. 1 -

 1 Fig. 1 -Nominal Scanning Law of Gaia. (Credits: ESA)

Fig. 2 -Fig. 3 -

 23 Fig. 2 -Number of NEAs and PHAs observed by Gaia with respect to the initial precession angle.

Fig. 5 -

 5 Fig. 5 -Example of geocentric distributions (α, δ) for the predicted positions on the 3th day after discovery with only Gaia observations (left 1σ uncertainty) and on the 4th day with an additional geocentric observation (right 10σ uncertainty). The triangle represents the expected value.


J Berthier

Identification of Known SSO in CU4 Object Processing

published or not. The documents may come

Identification of Known SSO in CU4 Object Processing

Introduction

The identification of known solar system objects (SSO) that will be observed by Gaia is a key point of the solar system object processing pipeline (CU4.SSO). It aims to associate the provisional tag assigned to observations of probably solar system objects to already known targets. At the time Gaia flies, it can be estimated that about 600,000 solar system objects (mainly small solar system objects) will be known and characterized by an orbit accurate enough to make their identification almost certain.

Context of the identification process

SSO, as well as non-single stars and extended objects, cannot be processed by the same pipeline than the stellar objects, mainly because:

-SSO have a proper motion and need a dedicated astrometric solution -SSO can be extended objects and need a dedicated photometric analysis -SSO are not observed at the same place at each transit so that we need to link up observations After being processed by CU3/IDT, a source is either successfully matched to an existing stellar object (i.e. all celestial object which is not SSO) or remains unmatched. In that case, a second processing is performed by CU4/SSO/DU452 to try to identify the source as a known SSO and to assign it an identifier, at the level of short-time processing (1 day), and at the level of long-term processing (6 months). In both cases, the success or not of the identification process decides which treatment is applied or not to the analyzed transit.

Identification method

The problem

The main problem to cross-match sources with SSO in astronomical images resides in the apparent motion and the relatively high number of SSO. The first point implies that the ephemeris of all known objects have to be accurately computed at the epoch of the image to be able to cross-match the sources. The second point implies that we must reckon on a rather long computing time: if we need 1 millisecond to compute the ephemeris of one SSO, then this will take more than 540 seconds (more than 540000 SSO are known at that time) to identify only one source! This is not possible when thousands of sources have to be processed daily.

A solution

Being moving objects, the cross-matching of an observed source with a known SSO requires to compute the ephemeris of each of them at the time of observation because we don't know a priori where they are located in the solar system. To solve this issue, we adopted the solution to compute in advance the ephemerides of all the known SSO at tabulated epochs, and to store them in a dedicated database. The identification process is then reduced to a simple cone-search method, supplemented by the computation of accurate ephemeris of a limited set of bodies.

Fig. 1 -Healpix cutting of the celestial sphere

On the base of our experience with such problem [START_REF] Berthier | SkyBoT, a new VO service to identify Solar System objects[END_REF], we have chosen to use the mathematical structure of Healpix [START_REF] Górski | HEALPIX -a Framework for High Resolution Discretization, and Fast Analysis of Data Distributed on the Sphere[END_REF] to store the tabulated positions in function of time of all known SSO. The sphere is hierarchically tessellated into curvilinear quadrilaterals, with the lowest resolution partition comprised of 12 base pixels. The resolution of the tessellation is then increased by division of each pixel into four new ones. The figure 1 illustrates (clockwise from upper-left to bottom-left) the resolution increase by three from the base level (i.e. the sphere is partitioned, respectively, into 12, 48, 192, and 768 pixels). The properties of Healpix implies that areas of all pixels at a given resolution are identical, and that pixels are distributed on lines of constant latitude.

These properties of Healpix makes it very efficient to store the coordinates of SSO because it supports a suitable discretization of functions on a sphere to different levels of resolution, so that it facilitates fast and accurate statistical analysis. We have chosen to use a grid resolution of k = 9 (Nside = 512), so that the sphere is cut in 3145728 pixels of angular size 6.87 arcmin.That ensures us that only very few SSO may be selected at once before crossmatching, and thus that ensures a short computing time.

Conclusion

This work is still in progress, and we don't have numbers to quantify the success and the accuracy of the identification process. But to ensure a high level of success, we know that we need high accuracy orbital elements. That means that, at the time Gaia will fly, we will need complementary astrometric observations of i) unmatched sources in order to confirm or not that it is a SSO, and ii) ambiguous cross-match to improve the accuracy of orbital elements and then, perhaps, properly identify the SSO.


W Thuillot
email: william.thuillot@imcce.fr

Objectives and Management of the Gaia-FUN-SSO Network

Introduction

Ground-based observations in addition to observations by space missions can be a very efficient way to get complementary data, to go beyond the primitive objectives of this mission and actually to increase the scientific return of the mission. This strategy is used in the frame of several missions, for example when a ground-based network is required to get a large coverage of observation, for a fast verification of an event or its follow-up with instruments unavailable on board. For example, the Corot space observatory for asterosismology and detection and study of exoplanets, or the future Swom space mission for the detection and study of Gamma Ray Bursts, apply this strategy. The space-ground synergy is also an efficient strategy for the Gaia mission and in particular for the Solar System Object science [START_REF] Thuillot | Complementary ground-based observations for Solar System applications[END_REF]. Several ground-based activities are being organized in the frame of the Gaia mission: the GBOT (Ground based Optical Tracking of Gaia, see Altmann et al. ibid) is a specific observing program of the probe itself; observing sites are also foreseen in a Science Alert Network for various science alerts mainly dedicated to astrophysical objects, photometry and spectroscopy; the Gaia-FUN-SSO program, presented here, is focused on an astrometric follow-up of specific Solar System objects.

The Gaia Follow-Up Network for Solar System Objects

Objectives

During its mission, Gaia will observe many Solar System objects and we can easily foresee that several interesting detections will be done during this five year mission. One of the important specificities of its observations is that they could be done at rather low solar elongation, 45 degrees, therefore detections of inner earth asteroids (IEAs, or Atens) could be performed. Detection of some new NEAs at bigger solar elongation could also be done. In these cases, due to the motion of the objects and perhaps to the limiting magnitude, the scanning law of Gaia will restrict the orbit determination to be founded on a very small number of astrometric measurements. In that case, only a ground-based network can avoid the loose of the asteroid and can permit to build an orbital modeling, based on enough astrometric measurements even at a lower accuracy than Gaia, in such a way that future observations remain possible. This is the primary objectives of the Gaia Follow-up Network for Solar System Objects.

In addition to the improvement of the astrometric data used for the orbital modeling of some specific objects, we can also think that some peculiar objects such as new comets, or even asteroids with cometary activity, could be detected. Due to the limiting factors of the Gaia observing method, ground-based networks can be very helpful in order to get fast more information upon the physical characteristics of these objects. Moreover, in particular during the first months of the mission, many unpredictable detections can arise, but the filtering parameters of the data processing will have to be tested and tuned and an important task will be to discriminate these detections, in particular for the detection of moving objects. Groundbased networks, such as the Gaia-FUN-SSO network, will be sollicitated for a contribution to this task.

Observing sites and instruments

In order to be efficient in case of the trigger of an alert by the Gaia data processing system, the network must have a large geographical coverage. This is why several observing stations have been contacted for a participation to this project. Almost 25 observing sites are candidate to be members of the network (cf. figure 1). Alerts from the Gaia data processing system can be received between 24 and 48 hours after detection. Difficulties may arise for the observation of peculiar objects: fast moving objects, faint objects, NEAs close to Earth therefore with strong parallax effect…Thus, even if some small instruments (smaller than 0.6 m) can be useful in this network, some telescopes with larger diameters, large field of view and some with sensitive detectors must be included in it. Furthermore, robotic telescopes could be very efficient instruments for such observations on alert.

The figure 2 shows a histogram of the telescope diameters according to the preliminary information received from the candidate's sites. Among these telescopes, 3 are Schmidt telescopes and 12 are robotic telescopes. Note that these observing sites and instruments will be subsequently much precisely determined thanks to a census organized at the time of this workshop.

Fig. 2 -Histogram of the telescope diameters of the candidate sites of the network

Organization

As described in figure 3, the Gaia-FUN-SO network will be composed with a Central Node (CN) and several Observing Sites (OS). The role of the CN will be the coordination of the network, the preparation of ephemerides and the dissemination of the data and results. The alerts related to the Solar System Objects will be received about 24 to 48 hours after the detection by Gaia. The rough format of the data, issued from the Gaia data processing system, will have to be transformed in ephemerides or in celestial coordinates of zone of interest for ground-based detection (depending on the ability to build ephemerides). Once a detection is done in an OS, astrometric measurements must be done and send to the CN. Improved ephemerides can then be done by the CN and disseminated in the network.

At the end of the observing process by the Gaia-FUN-SSO network, the observed coordinates of a newly detected SSO, or improved astrometric data for known objects, will be sent to the Minor planet Center. This will be the only way to inject new SSO data issued from the network in the auxiliary database used by the Gaia data processing for subsequent identification of SSO objects. A possible channel can also be used, to send these data to the ESA Space Situational Awareness programme when the SSO will be Near-Earth Objects.

How to be ready for the observations

For the next months we can foresee several important actions in order to have the Gaia-FUN-SSO in a good shape for the observations on alert as soon as the Gaia probe will be operating after the launch in spring 2013:

• For the Central Node: -some new observing sites could be get in touch in order to complete the geographical coverage of the network.

-more information are necessary for a better definition of the data format of the inputs/outputs and prepare the data useful for observing -a wiki will be set up to facilitate the exchange of information and tools -guidelines and recommendations will be posted on the wiki -an online agreement document will define the reciprocal commitments -tests and simulation of alerts will be organized -local organization must be set up for astrometric observations on alert (observers, accurate timescale in UT,…) and be ready to participate to observing tests almost one year before the launch -OS will have to agree the reciprocal commitments and to register on the wiki in order to access the data -OS will have to describe the site and instrument characteristics on the wiki

Conclusion

This will be probably the first time that coordinated ground-based measurement will improve, or even to secure, space astrometry measurements. In that sense, the Gaia-FUN-SSO network will be a fine opportunity to get benefit from the ground-space synergy and an important mean to improve the scientific return of the Gaia mission for the Solar System Objects science.

Fig. 1 -

 1 Fig. 1 -Geographical location of observing sites contacted to participate to Gaia-FUN-SSO

Fig. 3 -

 3 Fig. 3 -Scheme of processing of alerts on Solar System Objects