Francesco Topputo
email: francesco.topputo@polimi.it

Franco Bernelli-Zazzera

Earth-Mars Transfers with Ballistic Escape and Low-Thrust Capture ⋆

In this paper novel Earth-Mars transfers are presented. These transfers exploit the natural dynamics of n-body models as well as the high specific impulse typical of low-thrust systems. The Moon-perturbed version of the Sun-Earth problem is introduced to design ballistic escape orbits performing lunar gravity assists. The ballistic capture is designed in the Sun-Mars system where special attainable sets are defined and used to handle the low-thrust control. The complete trajectory is optimized in the full n-body problem which takes into account planets' orbital inclinations and eccentricities. Accurate, efficient solutions with reasonable flight times are presented and compared with known results.

Introduction

Low energy transfers outperform classic patched-conics orbits in terms of propellant mass [START_REF] Belbruno | Capture Dynamics and Chaotic Motions in Celestial Mechanics: With Applications to the Construction of Low Energy Transfers[END_REF]. The natural dynamics embedded in the n-body problems is exploited in these transfers. In particular, the ballistic capture mechanism avoids having hyperbolic excess velocities, so reducing the cost needed to insert the spacecraft into a final, stable orbit about the arrival planet [START_REF] Belbruno | Sun-Perturbed Earth-to-Moon Transfers with Ballistic Capture[END_REF]. Low energy transfers can also be thought under the perspective of Lagrangian point orbits and their invariant manifolds [START_REF] Belbruno | The Dynamical Mechanism of Ballistic Lunar Capture Transfers in the Four-Body Problem from the Perspective of Invariant Manifolds and Hill's Regions[END_REF][START_REF] Lo | Low Energy Interplanetary Transfers using the Invarian Manifolds of L1, L2, and Halo Orbits[END_REF][START_REF] Koon | Heteroclinic Connections between Periodic Orbits and Resonance Transitions in Celestial Mechanics[END_REF][START_REF] Gómez | Invariant Manifolds, the Spatial Three-Body Problem and Space Mission Design[END_REF][START_REF] Koon | Low Energy Transfer to the Moon[END_REF][START_REF] Gómez | Libration Point Orbits: A Survey from the Dynamical Point of View[END_REF][START_REF] Koon | Constructing a Low Energy Transfer between Jovian Moons[END_REF][START_REF] Marsden | New Methods in Celestial Mechanics and Mission Design[END_REF][START_REF] Topputo | Low Energy Interplanetary Transfers Exploiting Invariant Manifolds of the Restricted Three-Body Problem[END_REF]. In this case, the structure of the phase space about the collinear points of the restricted three-body problem is used to define efficient coast arcs. Trajectories obtained adopting two distinct three-body systems are matched together to define the transfer orbit. This technique is labeled "patched restricted three-body problems approximation" and represents a sophisticated update to the patched-conics method. Ballistic escape can be used besides ballistic capture to further enhance the trajectory performances. This is a mechanism equivalent to ballistic capture, and it is usually defined in a four-body model [START_REF] Belbruno | Examples of the Nonlinear Dynamics of Ballistic Capture and Escape in the Earth-Moon System[END_REF][START_REF] Topputo | Resonant Motion, Ballistic Escape, and their Applications in Astrodynamics[END_REF].

The chemical, patched restricted three-body problems approximation requires that the manifolds of the two systems intersect in the configuration space. This occurs in the Sun-Earth-Moon scenario (in constructing Earth-Moon transfers) as well as in transfers between outer planets and their moons [START_REF] Lo | Low Energy Interplanetary Transfers using the Invarian Manifolds of L1, L2, and Halo Orbits[END_REF][START_REF] Gómez | Invariant Manifolds, the Spatial Three-Body Problem and Space Mission Design[END_REF][START_REF] Koon | Low Energy Transfer to the Moon[END_REF][START_REF] Topputo | A Hybrid Optimization of the Low Energy Interplanetary Transfers Associated to the Invariant Manifolds of the Restricted Three-Body Problem[END_REF]. No intersection exists among manifolds of inner planets. A method to design Earth-Mars transfers exploiting the Lagrangian point dynamics has been formulated in [START_REF] Topputo | Low Energy Interplanetary Transfers Exploiting Invariant Manifolds of the Restricted Three-Body Problem[END_REF]; this is called "patched conic-manifolds" method. All these methods are based on instantaneous velocity changes needed to connect the manifolds of two systems in the phase space. Low-thrust propulsion is proposed in this paper to further improve the performances of low energy transfers. As existing methods rely on impulsive maneuvers, the design strategy needs to be modified to accommodate low-thrust control. This is done by defining special attainable sets. In short, an attainable set is a collection of low-thrust orbits that are propagated from a set of admissible initial conditions, with a specified guidance law and for a specified time. As the patched threebody problems approximation requires the intersection of the manifolds, and this condition is not verified in the Earth-Mars context [START_REF] Topputo | Low Energy Interplanetary Transfers Exploiting Invariant Manifolds of the Restricted Three-Body Problem[END_REF][START_REF] Ren | Two Mechanism of Natural Transport in the Solar System[END_REF], the idea behind our approach is to replace invariant manifolds with attainable sets, and to manipulate the latter in the same way the manifolds are used to define space trajectories. Low-energy, low-thrust Earth-Mars transfers are so formulated. Not only the intrinsic dynamics of n-body problems is exploited, but also the high specific impulse of low-thrust systems is utilized in order to define efficient trajectories. These transfers are made up by a ballistic escape portion and a low-thrust capture. Ballistic escape is defined in the Moon-perturbed Sun-Earth model; low-thrust capture is instead defined in the Sun-Mars problem. These two pieces are joined and optimized in the controlled n-body problem. In this model, the spacecraft's low-thrust propulsion as well as the planet's gravitational attractions are taken into account. The low-thrust represents the control term as it is used to adjust the natural flow of the n-body equations of motion. This paper follows previous works by the same authors aimed at combining dynamical system theory and optimal control problems to design efficient space trajectories [START_REF] Topputo | Low-Thrust Non-Keplerian Orbits: Analysis, Design, and Control[END_REF][START_REF] Mingotti | Trajectory Design and Optimization in Highly Nonlinear Astrodynamics[END_REF].

The literature on low-thrust, n-body trajectories is vast. A spiral arc is matched to a Moon transit orbit in [START_REF] Belbruno | Lunar Capture Orbits, a Method of Constructing Earth-Moon Trajectories and the Lunar GAS Mission[END_REF] (this concept has been later implemented in ESA's SMART-1 mission [START_REF] Schoenmaekers | SMART-1: With Solar Electric Propulsion to the Moon[END_REF]). The use of invariant manifolds as first guess to initiate low-thrust optimization is described in [START_REF] Anderson | Role of Invariant Manifolds in Low-Thrust Trajectory Design[END_REF]. Capture and escape orbits have been obtained with sophisticated optimization algorithms in [START_REF] Whiffen | Application of the SDC Optimal Control Algorithm to Low-Thrust Escape and Capture Trajectory Optimization[END_REF]. Low-thrust propulsion has been used within the restricted three-body problem to design both interplanetary transfers [START_REF] Dellnitz | On Target for Venus -Set Oriented Computation of Energy Efficient Low Thrust Trajectories[END_REF][START_REF] Dellnitz | Set Oriented Approximation of Invariant Manifolds: Review of Concepts for Astrodynamical Problems[END_REF][START_REF] Pergola | Earth-Mars Halo to Halo Low Thrust Manifold Transfers[END_REF] and transfers to the Moon [START_REF] Mingotti | Low-Energy, Low-Thrust Transfers to the Moon[END_REF][START_REF] Mingotti | Numerical Methods to Design Low-Energy, Low-Thrust Sun-Perturbed Transfers to the Moon[END_REF]. Low-thrust, stable-manifold transfers to halo orbits are also shown in [START_REF] Starchville | Optimal Low-Thrust Trajectories to Earth-Moon L2 Halo Orbits (Circular Problem)[END_REF][START_REF] Starchville | Optimal Low-Thrust Transfers to Halo Orbits about the L2 Libration Point in the Earth-Moon System (Elliptical Problem)[END_REF][START_REF] Sukhanov | Low Thrust Transfer to Sun-Earth L1 and L2 Points with a Constraint on the Thrust Direction[END_REF][START_REF] Senent | Low-Thrust Variable-Specific-Impulse Transfers and Guidance to Unstable Periodic Orbits[END_REF][START_REF] Mingotti | Combined Optimal Low-Thrust and Stable-Manifold Trajectories to the Earth-Moon Halo Orbits[END_REF][START_REF] Howell | Low-Thrust Transfers in the Earth-Moon System Including Applications to Libration Point Orbits[END_REF][START_REF] Martin | Space Manifold Dynamics, chapter Optimal Low-Thrust Trajectories to the Interior Earth-Moon Lagrange Point[END_REF]. Lunar gravity assists at departure in the frame of Earth-Mars transfers have been proposed in [START_REF] Gil-Fernández | ExoMars Alternative Escape Trajectories with Soyuz/Fregat[END_REF].

Summary of the Approach

The approach is briefly sketched for the sake of clarity (details are given throughout the paper). To construct a chemical, low-energy transfer between the Earth and Mars, it is required that the invariant manifolds of the two restricted three-body problems (Sun-Earth and Sun-Mars models) intersect at least in the position space. Given the problem geometry, the unstable manifold of the L 2 orbits is considered in the Sun-Earth problem (W u L2 SE), and the stable manifold of the L 1 orbits is considered in the Sun-Mars problem (W s L1 SM). If these intersected, an Earth-Mars low energy transfer with at most one deep-space maneuver would exist. Unfortunately, these two objects do not intersect, and this can be viewed on a common surface of section in Fig. 1(a) where the two manifolds are reported in the Sun-Earth rotating frame. A first attempt to get the two sets closer consists in the introduction of a lunar gravity assist at departure. The associated ballistic escape orbits, defined in the Moon-perturbed Sun-Earth problem, are represented by the dashed line in Fig. 1(b). Although the lunar gravity assist moves the set of ballistic escape orbits towards the Sun-Mars manifold, it is not enough to perform the intersection. Such intersection can be achieved when low-thrust propulsion is introduced. The low-thrust capture set in Fig. 1(c) is obtained in the controlled Sun-Mars problem with tangential thrust as detailed in Section 3.2. The intersection between this set and the ballistic escape orbits defines the transfer point.

(Note that the low-thrust capture set intersects also the L 2 orbits unstable manifold; nevertheless, considering its intersection with orbits flying-by the Moon produces solutions requiring less propellant). This point uniquely identifies a first guess transfer orbit which is later optimized in the controlled, restricted n-body problem.

The paper is organized as follows. In Section 2 the problem is stated and some background notions to design ballistic capture and escape orbits using the Lagrangian points dynamics are recalled. In the same section the Moon-perturbed Sun-Earth model is presented. Section 3 introduces the low-thrust propulsion into the restricted three-body problem and defines the attainable sets. In Section 4 the design strategy is formulated: ballistic escape orbits and Mars low-thrust capture trajectories are matched together. The complete transfer optimization is defined in Section 5 and the obtained results are discussed in Section 6. Concluding remarks are given in Section 7.

Background

Statement of the Problem

In the linked-conics method used in interplanetary trajectory design, the spacecraft motion is studied outside of the spheres of influence of the planets, and heliocentric rendez-vous transfers are treated. In the patched-conics approximation, the two-body motion about the planets is studied, and the associated solutions are connected using the concept of sphere of influence [START_REF] Kemble | Interplanetary Mission Analysis and Design[END_REF]. Interplanetary trajectories considered in this work connect an orbit around the Earth with an orbit around Mars. Planet-centered frameworks are indeed considered to exploit the three-and four-body dynamics governing ballistic capture and escape. This can be thought as a refinement of both linked-and patched-conics methods.

The spacecraft is assumed to be initially on a circular parking orbit around the Earth at a given altitude h E . An initial impulsive maneuver, whose magnitude is ∆v E , is provided to place the spacecraft on a trajectory that escapes from the Earth. The energy level of this Earthescaping orbit is lower than that associated to two-body trans-Mars injection hyperbolas. In addition, this trajectory exploits the gravitational attractions of the Sun, Earth, and Moon. For these reasons this phase is called ballistic escape.

After the launch, the spacecraft can only rely on its own low-thrust propulsion to rendezvous with Mars and to descend down to a stable orbit around it. This phase is called low-thrust capture as the Sun-Mars interaction is exploited (besides the low-thrust) to reach the final orbit. The final orbit has eccentricity, e, and periapsis (or apoapsis), r p (or r a ), prescribed by mission requirements. The transfer is assumed to terminate when the spacecraft reaches the periapsis of the final orbit. Three out of the four parameters needed to specify an orbit in a planar context are given. The fourth (orbit orientation, ω) is determined in the trajectory optimization.

As both chemical and low-thrust propulsions are considered, our approach may also be labelled as producing hybrid propulsion transfers.

The Planar Circular Restricted Three-Body Problem

The motion of the spacecraft, P 3 , of mass m 3 , is studied in the gravitational field generated by two primaries, P 1 , P 2 , of masses m 1 , m 2 , respectively, assumed to move in circular motion about their common center of mass (Fig. 2(a)). It is assumed that P 3 moves in the same plane of P 1 , P 2 under the equations [START_REF] Szebehely | Theory of Orbits: The Restricted Problem of Three Bodies[END_REF] ẍ

-2 ẏ = ∂Ω ∂x , ÿ + 2 ẋ = ∂Ω ∂y , (1)
where the auxiliary function is

Ω(x, y, µ) = 1 2 (x 2 + y 2 ) + 1 -µ r 1 + µ r 2 + 1 2 µ(1 -µ), (2)
and µ = m 2 /(m 1 + m 2 ) is the mass parameter of the three-body problem. Eqs. [START_REF] Belbruno | Capture Dynamics and Chaotic Motions in Celestial Mechanics: With Applications to the Construction of Low Energy Transfers[END_REF] are written in a barycentric rotating frame with nondimensional units: the angular velocity of P 1 , P 2 , their distance, and the sum of their masses are all set to the unit value. Thus, P 1 , P 2 have scaled masses 1 -µ, µ, and are located at (-µ, 0), (1 -µ, 0), respectively. The distances in Eq. ( 2) are therefore

r 2 1 = (x + µ) 2 + y 2 , r 2 2 = (x + µ -1) 2 + y 2 .
(3) For fixed µ, the energy of P 3 is represented by the Jacobi integral which reads

J(x, y, ẋ, ẏ) = 2Ω(x, y) -( ẋ2 + ẏ2 ), (4)
and, for a given energy C, it defines a three-dimensional manifold

J (C) = {(x, y, ẋ, ẏ) ∈ R 4 | J(x, y, ẋ, ẏ) -C = 0}. ( 5
)
The projection of J on the configuration space (x, y) defines the Hill's curves bounding the allowed and forbidden regions of motion associated with prescribed values of C. The restricted three-body problem (RTBP) has five well-known equilibrium points, L j , whose energy is C j , j = 1, . . . , 5. This study considers the dynamics of the two collinear points L 1 and L 2 , that behave, linearly, as a saddle × center. There exists a family of retrograde Lyapunov orbits around L 1 , L 2 , and two-dimensional stable and unstable manifolds emanating from them [START_REF] Szebehely | Theory of Orbits: The Restricted Problem of Three Bodies[END_REF][START_REF] Conley | Low Energy Transit Orbits in the Restricted Three-Body Problem[END_REF][START_REF] Llibre | Transversality of the Invariant Manifolds Associated to the Lyapunov Family of Periodic Orbits Near L2 in the Restricted Three-Body Problem[END_REF].

The RTBP is used alternatively to model the spacecraft motion in the Sun-Earth (SE) and Sun-Mars (SM) systems, whose mass parameters are µ SE = 3.0034 × 10 -6 and µ SM = 3.2268 × 10 -7 , respectively. As for the SE model, the generic periodic orbit about Lj, j = 1, 2, is referred to as γ j , whereas its stable and unstable manifolds are labeled W s (γ j ), W u (γ j ), respectively. In the SM model, the periodic orbits are called λ j and their manifolds are W s (λ j ), W u (λ j ), j = 1, 2. Fig. 2 Three-and four-body problems described in Sections 2.2 and 2.3.

The Moon-perturbed Sun-Earth Model

When the RTBP is used to model the motion in the Sun-Earth system, the Moon is considered with the Earth as a whole. In this case, P 2 is the Earth-Moon barycenter and m 2 considers the masses of the Earth and the Moon. Merging the Earth and the Moon is a good approximation, and in general works well. However, it would be desirable to explicitly exploit the presence of the Moon, at least in its neighborhood. Ballistic escape trajectories may take advantage of Moon gravity assists to reduce the trans-Mars injection cost [START_REF] Gil-Fernández | ExoMars Alternative Escape Trajectories with Soyuz/Fregat[END_REF]. Therefore, a Moon-perturbed Sun-Earth problem is formulated. This model preserves the structure of the SE RTBP, and considers the lunar perturbation in the same way the Sun-perturbed Earth-Moon model is formulated in literature [START_REF] Simó | The Bicircular Model near the Triangular Libration Points of the RTBP[END_REF]. Some assumptions are made:

-2 ẏ = ∂Ω M ∂x , ÿ + 2 ẋ = ∂Ω M ∂y , θ = ω M (6)
with

Ω M (x, y, θ) = Ω(x, y, µ SE ) + m M r M - m M ρ 2 M (x cos θ + y sin θ). (7)
The last term in Eq. ( 7) considers the fact that the Moon does not orbit around a fixed Earth, but both the Earth and the Moon rotate around their common center of mass. The dimensionless physical constants introduced to describe the Moon influence are in agreement with those of the SE model. These are derived starting from primitive values [START_REF] Williams | Planetary Fact Sheets[END_REF]. Thus, the scaled distance between the Moon and the Earth is ρ M = ρ 3 /l = 2.5721 × 10 -3 , where ρ 3 is the Earth-Moon distance and l is the Sun-Earth distance; the scaled mass of the Moon is

m M = m 3 /(m 1 + m 2 ) = 3.6942 × 10 -8
, where m 1 , m 2 , and m 3 are the masses of the Sun, the Earth, and the Moon, respectively; the angular velocity of the Moon is

ω M = ω M /n -1 = 1.2367 × 10 1
, where ω M and n are the angular velocities of the Earth-Moon and Sun-Earth systems, respectively [START_REF] Mingotti | Trajectory Design and Optimization in Highly Nonlinear Astrodynamics[END_REF].

The instantaneous location of the Moon is (1

-µ SE + ρ M cos θ, ρ M sin θ), such that the distance (a) W s (γ2) and W u (γ2). (b) W u (γ2).
Fig. 3 Stable and unstable manifolds W s (γ2), W u (γ2), and a sample ballistic escape orbit.

between P 3 and the Moon is

r 2 M = (x -1 + µ SE -ρ M cos θ) 2 + (y -ρ M sin θ) 2 . (8)
Equilibrium points, periodic orbits around them, their invariant manifolds, and the Jacobi integral disappear when Eqs. [START_REF] Gómez | Invariant Manifolds, the Spatial Three-Body Problem and Space Mission Design[END_REF] are considered. This also causes the loss of geometrical properties of the phase space in the libration points region: the separatrix role of the invariant manifolds [START_REF] Koon | Heteroclinic Connections between Periodic Orbits and Resonance Transitions in Celestial Mechanics[END_REF][START_REF] Gómez | Invariant Manifolds, the Spatial Three-Body Problem and Space Mission Design[END_REF] no longer applies. The reader can refer to [START_REF] Mingotti | Trajectory Design and Optimization in Highly Nonlinear Astrodynamics[END_REF] for the derivation of equations ( 6)-(8).

Ballistic Escape from the Earth

In the SE model, an energy value C SE C 2 is fixed such that γ 1 , γ 2 exist, and the Hill's regions are opened at both L 1 and L 2 . The periodic orbits and their invariant manifolds, W s,u (γ 1,2 ), can be computed with standard algorithms [START_REF] Gómez | Study of the Transfer from the Earth to a Halo Orbit around the Equilibrium Point L1[END_REF][START_REF] Parker | Practical Numerical Algorithms for Chaotic Systems[END_REF]. In the following, we construct a ballistic escape orbit exploiting W s (γ 2 ) and W u (γ 2 ) in the same way translunar orbits are obtained in Earth-Moon low energy transfers. The reader can refer to [START_REF] Mingotti | Low-Energy, Low-Thrust Transfers to the Moon[END_REF] for details.

Two surfaces of section are introduced to study the cuts of W s (γ 2 ), W u (γ 2 ) (Fig. 3). Section S A , making an angle ϕ A (clockwise) with the x-axis and passing through the Earth, is considered to cut W s (γ 2 ) (in Fig. 3(a), ϕ A = π/2). Section S B , inclined by ϕ B (clockwise) from the x-axis and passing through the Sun, is assumed for W u (γ 2 ) (in Fig. 3(b), ϕ B = π/4). Cutting W s (γ 2 ), W u (γ 2 ) with S A , S B produces the curves ∂Γ s 2 , ∂Γ u 2 which are diffeomorphic to circles [START_REF] Koon | Heteroclinic Connections between Periodic Orbits and Resonance Transitions in Celestial Mechanics[END_REF][START_REF] Gómez | Invariant Manifolds, the Spatial Three-Body Problem and Space Mission Design[END_REF]. (For some values of C SE and ϕ A , W s (γ 2 ) may experience close encounters or collisions with the Earth; in this case its section curve is no longer a circle). These cuts can be represented in (r 2 , ṙ2 ) and (r 1 , ṙ1 ) coordinates, respectively (in Fig. 4(b), ∂Γ s 2 is reported). Both sections represent two-dimensional maps for the flow of the RTBP. Indeed, any point on these sections uniquely defines an orbit. This property holds as J (C SE ) and S A,B lower the dimension of the phase space by two. By definition, points on ∂Γ s 2 originate orbits that asymptotically approach γ 2 in forward time. Points inside ∂Γ s 2 generate transit orbits that pass from the Earth region to the exterior region, whereas points outside Γ s 2 correspond to nontransit orbits (the manifolds act as separatrices for the states of motion [START_REF] Koon | Heteroclinic Connections between Periodic Orbits and Resonance Transitions in Celestial Mechanics[END_REF][START_REF] Conley | Low Energy Transit Orbits in the Restricted Three-Body Problem[END_REF][START_REF] Llibre | Transversality of the Invariant Manifolds Associated to the Lyapunov Family of Periodic Orbits Near L2 in the Restricted Three-Body Problem[END_REF]). Fig. 4 Ballistic escape trajectory performed with a tangential ∆vE maneuver and its associated section point P .

Ballistic escape takes place on orbits inside both W s (γ 2 ) and W u (γ 2 ). Let Γ s 2 be the set of points in the (r 2 , ṙ2 )-plane that are enclosed by ∂Γ s 2 (see Fig.

l = {(r 2 , ṙ2 ) ∈ S A , (r 2 , ṙ2 ) ∈ Γ s 2 |r 2 = R E + h E } (9)
are ballistic escape orbits that intersect the initial parking orbit (R E is the radius of the Earth). This intersection happens only in the position space, as the initial parking orbit and the escape trajectory show two different energy levels. The pair {C SE , ϕ A } uniquely defines the curve ∂Γ s 2 on S A (C SE defines the orbit γ 2 ; ϕ A defines the surface S A to cut W s (γ 2 )). It can be shown that {C SE , ϕ A } may be suitably tuned to produce transit orbits tangent to the Earth-parking orbit [START_REF] Mingotti | Low-Energy, Low-Thrust Transfers to the Moon[END_REF]. Typical values of Jacobi constant range in the interval C SE ∈ [3.0001, 3.0002] (C 2 ≃ 3.0008 in the SE model); as for the surface of section, ϕ A ∈ [π/2, 3/2π]. With reference to Figure 4(a), these values guarantee that the stable manifold W s (γ 2 ), backward integrated, is tangent to the parking orbit; this yields tangential insertion maneuvers: the initial ∆v E is aligned with the velocity of the circular parking orbit. The search is therefore restricted to the points P ∈ S A defined by P = l ∩ l ′ , where l ′ is the set of points having zero radial velocity with respect to the Earth

l ′ = {(r 2 , ṙ2 ) ∈ S A , (r 2 , ṙ2 ) ∈ Γ s 2 | ṙ2 = 0}. ( 10
)
As at this stage a first guess solution is designed to be later optimized, orbits sufficiently close to P can also be considered. In particular, points P ′ ∈ S A are considered as well, such that |P ′ -P | ≤ ε, where ε is a certain prescribed distance. (Numerical experiment show that values of ε ranging from 10 -6 to 10 -5 guarantee convergence of subsequent optimizations). A number of P ′ points can be generated by varying ϕ A . These points, flown forward, generate orbits that are close to W s (γ 2 ) until the region about γ 2 is reached. From this point on, the orbits get close to W u (γ 2 ), and their intersection with S B is studied. The set labeled E SE , E SE ∈ S B , represents the set of orbits close to W u (γ 2 ) whose pre-image E -1 SE , E -1 SE ∈ S A , is made up by P ′ points. Trajectories defined on E -1 SE , E SE , are of interest, as they lead to ballistic escape orbits.

Ballistic Escape with Lunar Gravity Assist

The set E SE is obtained in the SE RTBP. When the set E SE is integrated in the Moon-perturbed SE model, its pre-image E -1 SE shows negligible differences with that obtained in the RTBP. However, there are some orbits in E SE that experience close encounters with the Moon. With the values of ϕ A given above, close encounters with the Moon take place at θ ∈ [-π/3, -π/6], where θ is the phase angle of the Moon in Eq. [START_REF] Gómez | Libration Point Orbits: A Survey from the Dynamical Point of View[END_REF]. For these orbits, the points P ′ ∈ S A have almost the same coordinates (r 2 , ṙ2 ) as before, but the tangential velocity reduces significantly. More specifically, when a lunar gravity assist is explicitly taken into account, the difference in energy level between the parking orbit and the orbit on E -1 SE is reduced. This yields a lower ∆v E needed to place the orbit on a trans-Martian transfer. On the other hand, the effect of the Moonperturbation is negligible when the spacecraft flies far from the Earth (the two-body model is already a suitable approximation of the problem, as discussed in [START_REF] Topputo | Low Energy Interplanetary Transfers Exploiting Invariant Manifolds of the Restricted Three-Body Problem[END_REF]). In Fig. 5(b) the set E SE is reported. In the RTBP, this set is supposed to be inside ∂Γ u 2 . When the orbits are integrated in the Moon-perturbed SE model, the lunar gravity assist let E SE to go outside ∂Γ u 2 .

Ballistic Capture at Mars

Ballistic capture at Mars is designed in analogy with ballistic escape from the Earth. To approach Mars from the interior, a capture via L 1 is considered. In the SM model, the energy level is restricted to values C SM C 1 such that λ 1 exists, and the Hill's regions are opened at L 1 . The manifold W s (λ 1 ) is computed until a certain surface of section is reached. Section S C , making an angle ϕ C (clockwise) with the x-axis and passing through the Sun, is considered to cut W s (λ 1 ) (in Fig. 5(a), ϕ C = π/4). The corresponding section curve, ∂Λ s 1 , is represented in (r 1 , ṙ1 ) coordinates. The set K SM = Λs 2 is defined, where Λs 1 ∈ S C is the set of points inside ∂Λ s

1 . The points belonging to K SM are the ones that lead to Mars ballistic capture. These indeed generate orbits inside W s (λ 1 ) that are the only ones that approach Mars from the interior.

The set K SM is defined on section S C in the SM model. However, it is possible to represent K SM on S B defined in the SE model through a transformation KSM = M(K SM ). The operator M maps states on S C (SM model) to states on S B (SE model). It is constructed in five steps: i) the states of the SM model are written in the inertial reference frame with origin at the Sun-Mars barycenter; ii) the scaled variables are transformed into physical coordinates; iii) the origin is moved to the Earth; iv) the variables are scaled considering the SE physical constants; v) the variables are reported into the Sun-Earth rotating frame. The same conversion is also applied to ∂Λ s 1 , in order to obtain ∂ Λs 1 = M(∂Λ s 1 ) on section S B from section S C . In Fig. 5(b) both KSM and ∂ Λs 1 are reported. Considering section S B alone, Earth-Mars low energy transfers could be defined by E SE ∩ KSM . If this intersection had not been empty, the manifolds of the two systems would have been joined by a single deep-space maneuver performed at the patching point. Unfortunately Fig. 5(b) demonstrates that this intersection does not happen, nor it occurs in short-scale times [START_REF] Ren | Two Mechanism of Natural Transport in the Solar System[END_REF]. (In [START_REF] Gladman | Delivery of Planetary Ejecta to Earth[END_REF] it is shown that these orbits may match in million years, which is not likely the case of real space missions). In [START_REF] Topputo | Low Energy Interplanetary Transfers Exploiting Invariant Manifolds of the Restricted Three-Body Problem[END_REF], a two-impulse strategy was proposed to match the two manifolds. In [START_REF] Pergola | Earth-Mars Halo to Halo Low Thrust Manifold Transfers[END_REF], a low-thrust halo-to-halo rendez-vous was obtained. In [START_REF] Nakamiya | Interplanetary Transfers between Halo Orbits: Connectivity between Escape and Capture Trajectories[END_REF], Earth-Mars halo-to-halo transfers are designed by applying impulsive maneuvers at the periapsis of the manifolds. In this work, low-thrust capture down to low-altitude orbits about Mars is proposed. This is achieved through the introduction of low-thrust propulsion and the definition of special attainable sets.

Low-Thrust Propulsion and Attainable Sets

The Controlled, Planar Circular Restricted Three-Body Problem

To model the motion of a massless particle P 3 under both the gravitational attractions of P 1 , P 2 , and the low-thrust propulsion, the controlled RTBP is introduced

ẍ -2 ẏ = ∂Ω ∂x + T x m , ÿ + 2 ẋ = ∂Ω ∂y + T y m , ṁ = - T I sp g 0 , (11)
where T = (T 2 x + T 2 y ) 1/2 is the thrust magnitude, I sp the specific impulse of the thruster, and g 0 the Earth gravitational acceleration at sea level. Continuous variations of the spacecraft mass, m, are taken into account through the last of Eqs. [START_REF] Topputo | Low Energy Interplanetary Transfers Exploiting Invariant Manifolds of the Restricted Three-Body Problem[END_REF]. This increases the system order by one, and causes a singularity when m → 0 (in addition to the well-known singularities arising when P 3 collides with P 1 or P 2 ).

The thrust law T(t) = (T x (t), T y (t)) T , t ∈ [t i , t f ], in Eqs. [START_REF] Topputo | Low Energy Interplanetary Transfers Exploiting Invariant Manifolds of the Restricted Three-Body Problem[END_REF] is not given like in [START_REF] Dellnitz | On Target for Venus -Set Oriented Computation of Energy Efficient Low Thrust Trajectories[END_REF][START_REF] Dellnitz | Set Oriented Approximation of Invariant Manifolds: Review of Concepts for Astrodynamical Problems[END_REF], but rather in this approach it represents an unknown that is found by solving an optimal control problem (t i , t f are the initial, final times, respectively). T is determined in such a way that a certain state is targeted and a certain objective function is minimized at the same time. However, at this stage the profile of T over time is assigned to build first guess solutions. Attainable sets can be defined under this assumption.

Definition of Attainable Sets

Let y i be a vector representing a generic initial state, y i = (x i , y i , ẋi , ẏi , m i ), and let φ T(τ ) (y i , t i ; t) be the flow of system of Eqs. (11) at time t starting from (y i , t i ) and considering the thrust profile T(τ ), τ ∈ [t i , t]. With this notation, it is possible to define the generic point of a low-thrust trajectory through

y(t) = φ T (y i , t i ; t), (12)
where T(t) is the thrust law assigned. The low-thrust orbit, at time t, can be expressed as

γ T (y i , t) = {φ T (y i , t i ; τ )|τ < t}, (13)
where the dependence on the initial state y i is kept. The attainable set, at time t, can be defined as

A T (t) = yi∈Y γ T (y i , t), ( 14
)
where Y is a domain of admissible initial conditions. Attainable set in Eq. ( 14) is associated with a generic Y; this set can be defined for the low-thrust capture at Mars (see Section 4). The thrust law in Eq. ( 12) is now tailored for the problem at hand. When the spacecraft flies in heliocentric orbit, the low-thrust propulsion is used to rendez-vous with Mars. In this context the thrust is aligned with the velocity vector expressed in Sun-centered inertial coordinates. This strategy increases the semimajor axis in a given time. When the spacecraft approaches Mars, the low-thrust is used to achieve planet capture, and therefore the velocity is aligned with the velocity expressed in the Sun-Mars rotating frame. This tangential thrust is opposite to the velocity to maximize the variation of Jacobi energy. This is desirable to close the Hill's curves and stabilize the spacecraft about Mars. The thrust law used to define attainable sets is therefore

T(t) = T (v + r ⊥ )/|v + r ⊥ |, t ∈ [t i , t rv ] T(t) = -T v/|v|, t ∈ [t rv , t pc ] ( 15
)
where v = ( ẋ, ẏ), r ⊥ = (-y, x), and t rv -t i , t pc -t rv are the durations of the rendez-vous and planet capture phases, respectively. The term v + r ⊥ is the velocity expressed in the heliocentric inertial frame [START_REF] Dellnitz | On Target for Venus -Set Oriented Computation of Energy Efficient Low Thrust Trajectories[END_REF]. It is worth mentioning that the assigned guidance law in Eq. ( 15) will be let free to vary in the subsequent optimization step. Thanks to the definition of A T (t), low-thrust propulsion can be incorporated in a threebody frame using the same methodology developed for the invariant manifolds. More specifically, invariant manifolds are replaced by attainable sets which are manipulated to find a transfer point on a suitable surface of section. The idea is to mimic the role played by invariant manifolds. This is explained below.

Constructing Earth-Mars Transfers with Ballistic Escape and Low-Thrust Capture

The Earth-Mars transfer is made up by a ballistic escape orbit followed by a low-thrust rendezvous and subsequent planet capture at Mars. The ballistic escape orbit is constructed in the SE model through the method explained in Section 2.4. Low-thrust rendez-vous and capture is instead obtained in the SM model with Eqs. [START_REF] Topputo | Low Energy Interplanetary Transfers Exploiting Invariant Manifolds of the Restricted Three-Body Problem[END_REF] and using attainable sets (Section 3). The preliminary solution is defined by patching together these two "building blocks". The only task left is to specialize the domain of admissible conditions Y in Eq. ( 14) needed to define attainable sets.

The transfer ends when the spacecraft reaches the periapsis of the final orbit around Mars. Eccentricity and periapsis (apoapsis) radius of this orbit are fixed. The final state (i.e., the periapsis point) is function of the argument of periapsis, y f = y f (ω M ). The domain of admissible final states is

Y M = {y f (ω M ) | ω M ∈ [0, 2π]}. ( 16
)
The attainable sets for low-thrust capture are integrated backward starting from y f with the control law in Eq. ( 15). Thus, for some t = (t rv -t i ) + (t pc -t rv ), the attainable set containing low-thrust rendez-vous and capture trajectories is Since the first part of the transfer is defined on E SE (Section 2.4), the transfer points, if any, that generate low-energy, low-thrust Earth-Mars transfers are defined by

A M T (-t) = y f ∈Y M γ T (y f (ω M ), -t). ( 17
)
T M -t = E SE ∩ ÃM T (-t), (18)
where ÃM T (-t) stands for A M T (-t) mapped into the SE model through the map M in Eq. (??). A sample low-thrust capture trajectory and the associated attainable set are reported in Fig. 6. As first guess solutions are being generated with Eq. ( 18) (to be later optimized in the n-body model), small discontinuities can be again tolerated when looking for the transfer point. This means that it is possible to consider two states such that y Ay E ≤ ε, where y E ∈ E SE , y A ∈ ÃM T (t), and ε is a prescribed tolerance. The greater ε is, the higher number of first guess solutions is found; however, ε should be kept sufficiently small to permit the convergence of the subsequent optimization step. Again, values of ε of about 10 -6 have been considered in this work.

5 Trajectory Optimization in the n-Body Problem

The Controlled, Restricted n-Body Problem

Once feasible first guess solutions are found, they are optimized in a n-body problem frame. Assuming that the trajectories of the planets are given, the controlled, restricted n-body problem describing the spacecraft motion is

Ẍ = j ∈ B G m j R 3 j (X j -X) + T X m , Ÿ = j ∈ B G m j R 3 j (Y j -Y ) + T Y m , Z = j ∈ B G m j R 3 j (Z j -Z) + T Z m , ṁ = - T I sp g 0 , (19)
where R = (X, Y, Z) and R j = (X j , Y j , Z j ) are the positions of both the spacecraft and the j-th planet, respectively, expressed in an inertial reference frame; R j = |R j -R|, G is the universal gravitational constant, and m j is the mass of the j-th planet. In Eqs. [START_REF] Schoenmaekers | SMART-1: With Solar Electric Propulsion to the Moon[END_REF], B is the set containing the celestial bodies of interest; i.e., B = {Sun, Earth, Moon, Mars}. The last of Eqs. ( 19) is needed to introduce the low-thrust control, and to take into account the variation of the spacecraft mass m. The analytical ephemeris model provides the positions of the primaries as functions of time, i.e., X j = X j (t), Y j = Y j (t), Z j = Z j (t). This model is an approximation of JPL ephemeris DE405, where the positions of the planets are given as third-order polynomial of the epoch. It has been shown that such model is quite accurate for preliminary trajectory design purposes [START_REF] Armellin | Gravity Assist Space Pruning based on Differential Algebra[END_REF].

The low-thrust version of the classic restricted n-body problem consists of a seventh-order system of differential equations which describe the spatial problem. The spacecraft is allowed to move in three dimensions and the real eccentricities and orbital inclinations of the planets are considered. This model is therefore more accurate than the three-/four-body problems used to derive the first guess solutions. Thus, in the optimization step, first guesses are both improved (from a performance index point of view) and refined (from a dynamical model point of view).

Optimal Control Problem Statement

The optimal control problem is divided into three different stages according to the formalism proposed in [START_REF] Betts | Survey of Numerical Methods for Trajectory Optimization[END_REF][START_REF] Betts | Practical Methods for Optimal Control Using Nonlinear Programming[END_REF]. This is because solving the ballistic escape, the heliocentric phase, and the low-thrust capture, all with Eqs. ( 19) is not efficient as the terms R j show variations of several orders of magnitude. The n-body problem equations of motion are therefore written in frames centered at the Earth, Sun, and Mars. The three stages in which the optimal control problem is subdivided are: departure from the Earth; heliocentric orbit; arrival at Mars. Let y = (x, y, z, ẋ, ẏ, ż, m) be a generic state. Without loosing any generality, the stage notation is not used for the sake of brevity.

The optimal control problem aims at finding the guidance law, T(t), t ∈ [t i , t f ], that minimizes the following performance index

J = ρ ∆v E + t f ti T (t) I sp g 0 dt, (20)
where ∆v E is the magnitude of the Earth-escape maneuver. The second contribution to the objective function is the propellant mass, m p , spent in the low-thrust phase. (This can be derived by integrating the last of Eqs. ( 19)). The parameter ρ is a weight quantity introduced to balance the two contributions in the objective function.

The left boundary condition has to constrain the initial state on a circular parking orbit of radius r i = R E + h E about the Earth with velocity perpendicular to the position vector (R E is the Earth's mean radius). In inertial Earth-centered coordinates, the initial boundary condition reads

ψ i (y i , t i ) := x 2 i + y 2 i + z 2 i = r 2 i , x i ẋi + y i ẏi + z i żi = 0. ( 21
)
Under these conditions, the magnitude of the trans-Mars injection maneuver is

∆v E = ẋ2 i + ẏ2 i + ż2 i - µ E r i ( 22
)
where µ E is the Earth's gravitational parameter (µ E = 3.986 × 10 5 km 3 /s 2 ). Analogously, the final state y f expressed in inertial Mars-centered coordinates has to verify the right boundary condition

ψ f (y f , t f ) :=          x 2 f + y 2 f + z 2 f = r 2 f , x f ẋf + y f ẏf + z f żf = 0 ẋ2 f + ẏ2 f + ż2 f = µ M (1 + e) r f (23)
where r f and e are the periapsis radius and the eccentricity of the final orbit about Mars, respectively; µ M is the gravitational parameter of Mars (µ M = 4.282 × 10 4 km 3 /s 2 ). In addition, the following path constraint is imposed

T (t) ≤ T max , (24)
to model the saturation of the low-thrust engine. Eqs. ( 20)-( 24) define the optimal control problem for the Earth-Mars transfer. This problem may be solved with a variety of methods. In this work we have faced it with direct transcription and multiple shooting.

Solution by Direct Transcription and Multiple Shooting

The optimal control problem is transcribed into a nonlinear programming problem by means of a direct approach [START_REF] Hargraves | Direct Trajectory Optimization Using Nonlinear Programming and Collocations[END_REF]. This method generally shows robustness and versatility, and does not require explicit derivation of the necessary conditions of optimality; its convergence to a final solutions is also less sensitive to variations of the first guess solutions [START_REF] Betts | Survey of Numerical Methods for Trajectory Optimization[END_REF]. More specifically, a multiple shooting scheme is implemented [START_REF] Enright | Discrete Approximations to Optimal Trajectories Using Direct Transcription and Nonlinear Programming[END_REF]. With this strategy, the n-body equations of motion are forward integrated within N -1 intervals in which [t i , t f ] is split. This is done assuming N points and constructing the mesh

t i = t 1 < • • • < t N = t f .
The solution is discretized over these N grid nodes; i.e, y j = y(t j ). The matching of position, velocity, and mass is imposed at the endpoints of the intervals in the form of defects as η j = ȳjy j+1 = 0, j = 1, . . . , N -1 [START_REF] Mingotti | Low-Energy, Low-Thrust Transfers to the Moon[END_REF] with ȳj = φ T(τ ) (y j , t j ; t j+1 ), τ ∈ [t j , t j+1 ]. This is done in each of the three stages in which the problem is divided, and matching of position, velocity, and mass is also imposed at their endpoints. To compute T(τ ) a second-level time discretization is implemented by splitting each of the N -1 intervals into M -1 subsegments. The control is discretized over the M subnodes; i.e., T j,k , j = 1, . . . , N , k = 1, . . . , M . A third-order spline interpolation is achieved by selecting M = 4. Initial and final time t 1 , t N , are included into the nonlinear programming variable, so allowing the optimization of variable-time transfers. The transcribed nonlinear programming problem finds the states and the controls at mesh points (y j and T j,k ) in the respect of Eqs. ( 21)-( 24) and minimizing the performance index (Eq. ( 20)). It is worth stressing that not only the initial low-thrust portion, but rather the whole transfer trajectory is discretized and optimized, so allowing the low-thrust to act also in regions preliminarily made up by coast arcs. To find accurate optimal solutions without excessively increasing the computational burden, an adaptive nonuniform time grid has been implemented. When the trajectory is close to either the Earth or Mars the grid is refined, whereas in the intermediate phase, where the Sun attraction mostly governs the motion of the spacecraft, a coarse grid is used. This is done by hand; the implementation of an automatic mesh refinement scheme would be beyond the scopes of the paper. The optimal solution found is assessed a posteriori by forward integrating the optimal initial condition using an eighth-order Runge-Kutta-Fehlberg scheme (tolerance set to 10 -12 ) by cubic interpolation of the discrete optimal control solution.

Optimized Transfer Solutions

The optimal transfers presented connect the following orbits a circular orbit around the Earth at an altitude of h E = 167 km; -a circular orbit around Mars at an altitude of h M = 10000 km.

The latter has been chosen for the sake of comparison (it corresponds to the arrival orbit of a known example where ballistic capture at Mars is studied [START_REF] Topputo | Low Energy Interplanetary Transfers Exploiting Invariant Manifolds of the Restricted Three-Body Problem[END_REF]), though the method is formulated to reach any orbit about Mars once eccentricity and periapsis (apoapsis) altitude are specified. The results are summarized in Table 1 where 'fg' is the first guess and 'sol' is the corresponding optimized solution. Last two rows represent the reference, impulsive solutions.

Table 1 is organized as follows. In the second column, ∆v E is the magnitude of the initial impulsive maneuver. This is a direct output of the optimization step and is calculated through Eq. ( 22). In the third column, ∆v M is the magnitude of all impulsive maneuvers needed to reach the final orbit around Mars (this applies to the second reference solutions only, where three maneuvers are considered). In the fourth column, f f is the propellant mass fraction needed for the rendez-vous and the low-thrust capture (this number does not take into account the initial impulsive maneuver). For the optimized transfers, f f is calculated as

f F = 1 m T M t f ti T (t) I sp g 0 dt, (26)
where m T M is the mass injected into the trans-Mars orbit (m T M = 1000 kg). For the reference solution, f f is calculated as

f f = 1 -exp - ∆v M I ht sp g 0 , (27)
where I ht sp = 300 s is the specific impulse of high-thrust engines. In the fifth column, f t represents the overall mass fraction necessary to carry out the transfer. For the optimized solutions, f t is calculated through

f t = m p m i = 1 -exp - ∆v E I ht sp g 0 + 1 m i t f ti T (t) I lt sp g 0 dt, (28)
where m i is the initial mass (calculated to inject m T M = 1000 kg with a ∆v E maneuver), and I lt sp = 3000 s is the specific impulse of the low-thrust engines. The transfer time, ∆t, is reported in the sixth column. A maximum available thrust of T max = 0.25 N has been considered in Eq. ( 24). Sol 2 is reported in Fig. 7.

Discussion

In Table 1, the two optimized solutions show a shorter flight time and a higher mass consumption than the corresponding first guesses. This is because the optimal control problem spreads the discontinuity at the transfer point P ; more propellant than the first guess is spent, and the flight time is shorten. In addition, sol 2 outperforms sol 1 in terms of both flight time and ∆v E . This is due to the fact that fg 2 is designed to take explicitly advantage of a lunar flyby (see Fig. 7(c)). This feature reduces the magnitude of ∆v E by 110 m/s and gives reasons for the introduction of the Moon-perturbed SE model presented in Section 2.3. Moreover, the flight time is reduced by shrewdly tuning the angles ϕ B , ϕ C that define the plane where T M -t = E SE ∩ ÃM T (-t) is defined. Optimized solutions offer lower overall mass fraction f t than reference solutions. This happens for two reasons. Firstly, I lt sp is one order of magnitude greater than I ht sp . Secondly, the first guess solutions exploit the dynamics of the three-/four-body problems in which they are designed, and the optimized solutions efficiently use the 5-body problem accordingly. The flight times lie between those of the reference transfers. From Fig. 7(e) it can be inferred that the thrust profile respects the saturation constraint described by Eq. ( 24). The control profile recalls an on-off structure, which is valid for both the rendez-vous and low-thrust capture phases. We expect that with arrival and departure orbits with nonzero eccentricity, the structure of the transfer would not be too different from the solutions presented in this study, though a shorter flight time and a lower propellant mass fraction could be achieved. As for the inclination of the departure and arrival orbits, the optimized solution may have nonzero inclination (±15 deg determined through numerical experiments) even though the first guess is planar. If polar orbits about Mars are of interest, the described method is still valid provided that the first guess is constructed starting from high inclined arrival orbits.

Conclusions

A method to incorporate low-thrust propulsion into standard invariant manifold technique to design interplanetary transfers has been presented in this paper. This is done through the definition of special attainable sets that are manipulated together with invariant manifolds. This procedure recalls the one used in the patched restricted three-body problems approximation, and uses attainable sets to fill the gap in systems with nonintersecting manifolds (Sun-Earth and Sun-Mars systems in this case). The proposed transfers are made up by three phases: a ballistic escape, a low-thrust planetary rendez-vous, and a low-thrust capture. The ballistic escape exploits both a lunar gravity assist and the Sun-Earth gravitational attractions. This is defined in a special Moon-perturbed Sun-Earth problem developed purposely. The method has been specialized to Earth-Mars transfers although it can be used in a broader context. Efficient solutions have been shown to support the validity of the presented approach. Fig. 7 The optimized transfer corresponding to sol 2 in Table 1.

  (a) Non-intersecting manifolds. (b) Effect of lunar gravity assist. (c) Effect of low-thrust capture.

Fig. 1

 1 Fig.1Summary of the approach. Given the two non-intersecting manifolds (Fig.1(a)) defined into two restricted three-body problems, the Moon-perturbed Sun-Earth problem is introduced to design ballistic escape orbits exploiting lunar gravity assist (Fig.1(b), dashed line). This set is then connected with the low-thrust capture orbits obtained in the controlled Sun-Mars problem (Fig.1(c)). The intersection between ballistic escape and low-thrust capture sets uniquely specifies a first guess solution which is later optimized in the controlled, restricted n-body problem.

  (a) Planar circular RTBP. (b) Moon-perturbed Sun-Earth problem.

  (i) the Sun, the Earth, and the Moon orbit in the same plane; (ii) the Sun and the Earth move on circular orbits around their center of mass; (iii) the Moon moves in a circular orbit around the Earth (Fig.2(b)). With these assumptions the model is not coherent but it catches basic dynamics of the restricted four-body problem (RFBP) as the primaries have low eccentricities (0.0167 and 0.0549 for the Earth and Moon, respectively) and the Moon is inclined on the ecliptic by 5 deg.The equations of motion of this RFBP are ẍ

  (a) Initial maneuver geometry. (b) ∂Γ s 2 , Γ s 2 , l, l ′ , and point P = l ∩ l ′ .

  4(b)). Points on Γ s 2 are of interest. More specifically, all points lying on

  (a) W s (λ1) and a sample capture orbit. (b) ESE and KSM sets.

Fig. 5

 5 Fig. 5 Stable manifold W s (λ1), its section curve ∂ Λs 1 , Mars-capture set KSM , and the Earth-escape set ESE. Note that if ESE and KSE intersected (i.e., ESE ∩ KSM = ∅), low energy Earth-Mars transfers with at most one deep-space maneuver would exist.

  (a) Sample low-thrust capture trajectory. (b) ESE and ÃM T (-t) sets.

Fig. 6 A

 6 Fig. 6 A first guess low-thrust capture at Mars and the transfer point T M -t = ESE ∩ ÃM T (-t). By comparing Fig. 5(b) with Fig. 6(b), it is evident that introducing low-thrust capture allows having an intersection (and therefore a transfer point) which was missing with ballistic capture.

  (a) First guess solution (SE reference frame). (b) Optimal trajectory (heliocentric frame). (c) Ballistic escape with lunar gravity assist (inertial Earth-centered frame). (d) Low-thrust capture (inertial Mars-centered frame). (e) Optimal guidance law. (f) Mass consumption trend.

Table 1

 1 Low-energy, low-thrust transfers to low-Mars orbits. First guesses and their optimized solutions are reported together with two impulsive reference solutions (H: Hohmann).Type ∆v E [m/s] ∆v M [m/s] f f [adim.] f t [adim.] ∆t [days]

	fg 1	3260	-	0.150	0.719	756
	fg 2	3150	-	0.150	0.709	553
	sol 1	3253	-	0.199	0.734	703
	sol 2	3141	-	0.200	0.724	496
	H	3620	1878	0.472	0.846	259
	[11]	3554	1915	0.478	0.844	823


Therese Tillin
email: t.tillin@imperial.ac.uk

Nita G Forouhi

Paul M Mckeigue

Southall And Brent REvisited: Cohort profile of SABRE, a UK population based comparison of cardiovascular disease and diabetes in people of European, Indian Asian and African Caribbean

Keywords: ethnicity, diabetes, cardiovascular disease

HAL is

Asians [START_REF] Mckeigue | Diabetes, hyperinsulinaemia and coronary risk factors in Bangladeshis in East London[END_REF] . This hypothesis was to be tested in the Southall study, a large population-based inter-ethnic cross sectional study of middle aged men and women performed between 1988 and 1990 [START_REF] Mckeigue | Relation of central obesity and insulin resistance with high diabetes prevalence and cardiovascular risk in South Asians[END_REF] .

A large number of migrants of Black African descent from the Caribbean and West Africa also arrived in the UK during the post-Second World War period, encouraged by the UK government in an effort to fill the many vacancies for key workers [START_REF] Trust | Radical Statistics Race Group[END_REF] . As these African Caribbean migrants were also reaching middle age in the 1980s, an intriguing paradox was emerging in the UK and had been noted

previously in similar populations in the USA and Caribbean. Despite high rates of diabetes, people of African and African Caribbean descent had relatively low CHD risks compared with White European ethnic origin populations [START_REF] Miller | Ethnicity and other characteristics predictive of coronary heart disease in a developing country -principal results of the St James survey, Trinidad[END_REF][START_REF] Tunstall Pedoe | Coronary heart attacks in East London[END_REF][START_REF] Opcs | Mortality and geography: a review in the mid-1980s[END_REF][START_REF] Keil | Coronary disease mortality and risk factors in black and white men. Results from the combined Charleston, SC, and Evans County, Georgia, heart studies[END_REF] . In stark contrast, stroke-related mortality was doubled and mortality from hypertensive disease was increased between four-fold (men) and seven-fold (women)

in Caribbean born migrants compared with the general UK population [START_REF] Opcs | Mortality and geography: a review in the mid-1980s[END_REF] . In the UK this latter was especially puzzling, as previous population based studies did not provide unequivocal evidence that blood pressure, or the prevalence of hypertension, were elevated in people of Black African descent 12- 14 . Exploration of the diabetes/CHD paradox, and definitive exploration of ethnic differences in resting and ambulatory blood pressure formed the basis of the Brent population based comparison of African Caribbean and European middle aged men and women [START_REF] Chaturvedi | Resting and ambulatory blood pressure differences in Afro-Caribbeans and Europeans[END_REF] .

This cross-sectional comparison followed the successful completion of the Southall study in 1990.

The Southall and Brent studies were performed to identical protocols by the same team, using the same equipment and hospital laboratories. The geographical areas of the baseline studies are contiguous and the two studies were conducted consecutively. To improve statistical power, the two Cross-sectional analysis of the Southall and Brent studies yielded important information on disease prevalence and associations (see below), but could only hint at causality. In addition, first generation migrants as a cohort are now reaching pensionable age, where disease burden is anticipated to be high, yet little data exist for this age group. In particular, heart failure, as a consequence of CHD, diabetes and hypertension, a rarity in middle aged individuals, now becomes especially prevalent, causing over a third of all deaths in older age [START_REF] Murdoch | Importance of heart failure as a cause of death. Changing contribution to overall mortality and coronary heart disease mortality in Scotland 1979-1992[END_REF] . Finally, interim mortality follow-up analyses of this cohort showed striking ethnic differences in the effects of dysglycaemia on cardiovascular disease mortality (see below 'What has it found?') adding urgency to the need for follow-up study of cardiometabolic health and the mechanisms underlying ethnic differences.

The Southall And Brent REvisited (SABRE) study was initiated in 2008 as a 20 year follow-up of the two baseline studies.

Ethics committee approval was obtained for both baseline studies and for SABRE.

What does the study cover and how has this changed?

Broadly, the baseline studies were set up to examine, in cross-section, the associations between components of the insulin resistance or 'metabolic' syndrome in each of the three ethnic groups and to determine how these associations might underlie the higher rates of coronary heart disease in Indian

Asians and the higher rates of hypertensive target organ damage and stroke in African Caribbeans.

The SABRE follow-up study (2008-2013) continues the focus on ethnic differences in cardiometabolic risk. We are quantifying cardiometabolic events during the follow-up period in both surviving and deceased participants using questionnaires to participants and primary care record review, and in survivors, measuring subclinical cardiovascular disease. We will address the following primary research questions:

• Is the greater risk of CHD in Indian Asians explained by a hyperglycaemia-associated adverse lipid and lipoprotein profile compared to Europeans?

• Is the greater risk of stroke and MRI detected subclinical cerebral infarct in African Caribbeans explained by hyperglycaemia associated loss of nocturnal blood pressure dipping compared to Europeans?

• Does subclinical circulatory disease differ by ethnicity, and if so, is this predicted and explained by differences in expression of the cardiometabolic syndrome? Who is in the sample?

The Southall study (1988-1990) was the first of the two baseline studies [START_REF] Mckeigue | Relation of central obesity and insulin resistance with high diabetes prevalence and cardiovascular risk in South Asians[END_REF] . A target sample of 3000 men aged 40-69 years of Indian Asian and European descent was chosen to have 90% power to detect at 5% significance a relative risk of 1.5 for prevalent CHD in Indian Asian compared with European men. The sample was assembled from two sources selected for the ethnic mix of the potential participants: industrial workforces and general practitioners' lists. Although it was not possible to assign ethnicity to all potential participants invited, the response rates were similar in the two ethnic groups in those participants in whom ethnicity could be assigned.

All South Asian and African Caribbean participants are first generation migrants to the UK.

Table 1 here

How often have they been followed up?

Since the baseline studies, participants have been flagged for mortality by the Office for National

What has been measured?

At baseline, participants were invited to attend clinics in local hospitals and health centres for a health check after an overnight fast. Before the examination participants were asked to complete a selfadministered questionnaire that included items on past medical history (including diabetes and the Rose angina questionnaire [START_REF] Rose | Chest pain questionnaire[END_REF] , medication, smoking, alcohol intake, physical activity, dietary information, and occupation, together with questions regarding current and childhood socioeconomic circumstances. Ethnicity was described by the interviewer based on appearance and parental origin.

Clinical measurements were performed to a standard protocol and are listed in Table 2. Baseline characteristics are detailed in Table 3.

In addition to the above, Brent study participants were invited to take part in substudies including echocardiography, retinal photography and 24 hour ambulatory blood pressure monitoring. (Table 2)

Tables 2 and3 2.

We offer home visits from a trained nurse for those who live locally, but who are unable or unwilling to attend the clinic. These home visits include anthropometric and blood pressure measurements, plus urine and fasting blood samples. As part of the informed consent procedure, participants can also choose to limit their participation to completion of the questionnaire and/or consent for medical record review.

Cardiovascular and diabetes-related events which have occurred since the baseline studies are identified from primary care medical record review and from participant questionnaire responses. In the feasibility study of 176 people, non-responders (n=60) tended to have worse baseline risk factor profiles than responders, although the differences were seldom significant and not consistently in one direction. Based on the feasibility study, we expect overall response rates to be 70%, 65% and 60% among European, Indian Asians, African Caribbeans respectively. After nearly 2 years into the follow-up study, among 2925 approached, 61% of Europeans, 56% of Indian Asians and 52% of African Caribbeans have agreed to participate. We expect these rates to be substantially boosted following a further attempt to revisit or trace those who were initially away or whose addresses were unknown, and most importantly, following home visits by NatCen interviewers in the north-west London area.

At the expected response rates (70%, 65%, 60%), SABRE will have at least 80% power at 5% significance to examine the effects of major risk factors for CHD and stroke separately and to examine the different magnitudes of effect of diabetes (detectable hazard ratios in whole study population: new CHD events (fatal and non-fatal): Indian Asians vs Europeans: 1. What has it found?

The baseline Southall study was the first to report the high prevalence of the insulin resistance or cardiometabolic syndrome in association with a striking tendency to central obesity in British Indian Asian migrants-these findings were common to diverse South Asian sub-groups such as Hindus, Sikhs and Muslims. These findings led us to suggest that insulin resistance might be the underlying cause of the increased susceptibility to coronary heart disease and stroke among Indian Asians [START_REF] Mckeigue | Relation of central obesity and insulin resistance with high diabetes prevalence and cardiovascular risk in South Asians[END_REF][START_REF] Mckeigue | Relationship of glucose intolerance and hyperinsulinaemia to body fat pattern in south Asians and Europeans[END_REF] The Brent study, on the other hand, reported that whilst African Caribbeans were as dysglycaemic, they were less hyperinsulinaemic than Indian Asians. They were not centrally obese and had favourable blood lipid profiles suggesting that the causes of diabetes in African Caribbeans may be different from those in Indian Asians and this might also explain why coronary heart disease mortality in African Caribbeans is low, despite the high prevalence of diabetes and hypertension [START_REF] Chaturvedi | Relationship of glucose intolerance to coronary risk in Afro-Caribbeans compared with Europeans[END_REF] . In addition, resting and ambulatory blood pressure recordings in a subgroup of Europeans and African

Caribbeans (n =319) demonstrated that the higher resting blood pressures in African Caribbean women may be enough to explain their excess stroke mortality, but that blood pressure did not explain the excess stroke rates in African Caribbean men. Smaller declines in nocturnal blood pressure may further contribute to the excess of hypertensive target organ damage in African Caribbeans [START_REF] Chaturvedi | Resting and ambulatory blood pressure differences in Afro-Caribbeans and Europeans[END_REF] .

Echocardiographic study demonstrated that African Caribbeans had greater left ventricular structural impairment than Europeans, and that remodelling patterns of the left ventricle in response to hypertension differed, with greater wall thickening and cavity narrowing compared to Europeans [START_REF] Chaturvedi | Echocardiographic measures of left ventricular structure and their relation with rest and ambulatory blood pressure in blacks and whites in the United Kingdom[END_REF] .

This pattern of remodelling in other studies is found to have more adverse implications for downstream mortality than patterns more commonly observed in people of European origin.

Analyses of cardiovascular mortality during the first 16 years of follow-up suggested that diabetes (defined retrospectively by World Health Organisation, 1999 criteria [START_REF]Definition, Diagnosis and Classification of Diabetes Mellitus and its Complications. Part 1: Diagnosis and Classification of Diabetes Mellitus[END_REF] ) exerts a remarkably 'toxic' effect with regard to both CHD and stroke mortality in people of Indian Asian origin [START_REF] Forouhi | Do known risk factors explain the higher coronary heart disease mortality in South Asian compared with European men? Prospective follow-up of the Southall and Brent studies, UK[END_REF] . In people of African Caribbean origin, diabetes 'toxicity' was particularly striking in association with stroke mortality. At 16 years of follow-up African Caribbeans with diabetes were six times more likely to die from stroke than comparable Europeans (incidence rate ratio adjusted for age, systolic blood pressure and smoking status: 6.7 (95% CI:1.4, 32.5, p =0.018) [START_REF] Tillin | The role of diabetes and components of the metabolic syndrome in stroke and coronary heart disease mortality in U.K. white and African-Caribbean populations[END_REF] . Further unpublished analyses demonstrated similar effects on mortality in both Indian Asians and African Caribbeans in association with fasting and post-challenge glucose in both univariate and multivariate analyses (unpublished data, Figure ). These interactions between ethnicity and dysglycaemia could not be explained by conventional risk factors [START_REF] Forouhi | Do known risk factors explain the higher coronary heart disease mortality in South Asian compared with European men? Prospective follow-up of the Southall and Brent studies, UK[END_REF][START_REF] Tillin | The role of diabetes and components of the metabolic syndrome in stroke and coronary heart disease mortality in U.K. white and African-Caribbean populations[END_REF] and unpublished data and were evident even in those whose diabetes had not been diagnosed before the first baseline study visit. (Figure) However, numbers of deaths at this stage of follow-up were small and it is apparent that there is an urgent need to assess non-fatal events and subclinical cardiovascular disease in association with dysglycaemia in our surviving cohort. and non-fatal events, SABRE will uniquely be able to inform ethnic specific thresholds for CVD risk factor intervention in the UK.

The weaknesses of the study lie mainly in the single follow-up at 20 years and in loss to follow-up.

However, we are conducting review of medical records together with participant recall of major cardiovascular events since the baseline studies. This complementary approach has been shown to provide a valid method of identifying morbid events during lengthy follow-up periods [START_REF] Walker | Validation of patient recall of doctor-diagnosed heart attack and stroke: a postal questionnaire and record review comparison[END_REF] . In addition we have full mortality follow-up since the baseline follow-up and review of medical records will complement death certificate information for deceased participants. Loss to clinical follow-up is likely to be substantial, particularly in older migrant populations as retirement often involves a return to the home country for protracted periods. However, our pilot study data and initial responses indicate that most European and Indian Asian participants are traceable to a UK address and willing to take part in the follow-up study. It is our experience that African Caribbeans tend to favour face to face contact over a postal approach for recruitment, hence intensive efforts will be made to trace and visit African Caribbean non-responders.

Can I get hold of the data? Where can I find out more?

Data will be stored by the UK Data Archive from 2014. We welcome proposals from 2014 onwards for collaborative study of baseline and follow-up data. Please contact Professor Nish Chaturvedi (n.chaturvedi@imperial.ac.uk) for further information. Please also visit the SABRE study website www.sabrestudy.org.uk .

F

  Southall And Brent REvisited: Cohort profile of SABRE, a UK population based comparison of cardiovascular disease and diabetes in people of European, Indian Asian and African Caribbean origins

  now been amalgamated. Together, the Southall and Brent studies form the basis of the largest UK population based tri-ethnic cohort.Southall And Brent REvisited (SABRE)(2008-2013)

•

  Do current thresholds for risk factor interventions need modifying for UK ethnic minority groups?

  Statistics and analyses of death certificate data have been performed. A pilot postal follow-up in 176 people randomly selected from both baseline studies was conducted in 2004. This feasibility grant application for follow-up of the entire study population. The current study (SABRE) is the first follow-up of survivors.

here

  In the SABRE follow-up, all participants currently living in England or Wales are invited by letter to attend the study clinic at St Mary's Hospital London following an overnight fast. Taxi transport is provided for local participants and travel costs are reimbursed for those who live outside the London area. Participants are asked to complete a detailed questionnaire regarding medical history, lifestyle and socioeconomic circumstances. They then undergo a series of comprehensive clinical measurements as detailed in Table

  What are the main strengths and weaknesses?This is the largest tri-ethnic UK cohort and now has a 20 year follow-up. The South Asian and African Caribbean participants form a unique group of British first-generation migrants -a precious resource. The baseline studies were performed to a strict protocol and have resulted in detailed phenotyping of the group in middle age. The participants are now of an age to be at maximal risk of cardiovascular events and current follow-up involves extensive cardiovascular assessment using state of the art techniques. Particular strengths lie in the range and detail of the follow-up measurements of subclinical disease. These include indicators of suboptimal heart function and structure. Added value will be gained from validation of 2D echocardiographic measurements against gold standard 3D techniques for assessment of cardiac volumes. Cognitive function and MRI assessment of subclinical cerebral changes will provide unique data in this older population. Quantification of body fat distribution using CT together with glucose tolerance testing and measurements of lipoprotein subfractions will inform understanding of the mechanisms underlying the marked ethnic differences in the manifestations of insulin resistance. Studies involving ethnic group comparisons not only increase understanding of genetic and lifestyle effects on disease risk, but also form important natural experiments where the marked ethnic differentials in clustering of risk factors enable elucidation of the relative roles of each in disease causation. Finally, using baseline risk factor associations with fatal

  Reminder letters are sent to non-responders after two weeks. If there is no response after a reminder, then trained interviewers from the National Centre for Social Research (NatCen) carry out home visits in the north-west London area, with the aim of recruiting participants for full clinical follow-up and/or completion of the questionnaire and/or consent for medical record review.As SABRE represents the first individual follow-up of surviving participants and is in the midrecruitment phase, data on attrition are limited. We used the NHS tracing system to trace the cohort in 2008. Currently, we are nearly 2 years into the follow-up study and have approached 2925 of 4127 surviving participants. We have identified 431 people who either do not have a valid contact address or who were away when we attempted to make contact. This represents 8% of Europeans, 11% of Indian Asians and 19% of African Caribbeans. Although 22% of survivors (38% of Europeans, 7% of Indian Asians and 8% of African Caribbeans) have moved to areas outside North West London, the majority of dispersion is to other areas of greater London or the surrounding counties within a 50 mile radius.

	What is attrition like?
	F o r
	R
	e v
	i e
	w
	O n l
	y

Medical record review is undertaken by trained members of the research team in local general practices and at Primary Care Trusts where records of deceased participants are stored. Where participants have moved outside the local area, their GP is asked to complete the same case report form.

Table 1 : Summary of Southall and Brent study recruitment (1988-1991) and response rates and survival to SABRE follow-up (commencing 2008)

 1

	o r
	R
	e v
	i e
	w
	O n l
	y

Table 2 . Southall and Brent baseline studies and SABRE follow-up measurements

 2

	Investigations/measurements	Baseline studies
		(1988-91)

Table 3 . Baseline Characteristics (1988-91) in the combined Southall and Brent study population

 3

	Men
	Means ±SD, medians (25th, 75th	European	Indian Asian	P value	African	P value
	percentiles)	men	men	(IA vs E)	Caribbean	(AC vs E)
					men
	Number	1787	1420		453
	Age	52.7±7.1	50.7±7.0	<0.001	53.5±5.8	0.035
	Height, cm	174.2±6.9	169.8±6.5	<0.001	171.9±6.7	<0.001
	Weight, kg	79.6±12.9	74.5±11.0	<0.001	74.5±10.3	0.010
	Waist circumference, cm	91.9±11.0	92.9±9.6	0.013	89.4±9.8	<0.001
	Hip circumference, cm	97.3±6.7	94.6±6.2	<0.001	94.9±6.4	<0.001
	Waist to hip ratio	0.94±0.07	0.98±0.06	<0.001	0.94±0.06	0.66
	Body mass Index, kg/m 2	26.2±3.9	25.8±3.3	0.052	26.3±3.3	0.24
	Systolic blood pressure, mm Hg	121 (112, 133) 123 (113, 136)	<0.001	126 (116, 137)	<0.001
	Diastolic blood pressure , mm Hg	77 (70, 84)	80 (74, 87)	<0.001	80 (73, 89)	<0.001
	Treated hypertension	150 (8%)	190 (13%)	<0.001	85 (19%)	<0.001
	Prior CHD	169 (9%)	130 (9%)	0.79	23 (5%)	0.003
	Prior stroke	19 (1%)	25 (2%)	0.088	12 (3%)	0.009
	Known diabetes	55 (3%)	185 (13%)	<0.001	46 (10%)	<0.001
	Diabetes by WHO 1999 criteria,	127 (7%)	311 (22%)	<0.001	83 (18%)	<0.001
	n(%) ‡
	Impaired fasting glucose(isolated) ‡	171 (10%)	121 (9%)	0.31	52 (12%)	0.21
	Impaired glucose tolerance(isolated or	50 (3%)	70 (5%)	0.002	44 (10%)	<0.001
	both IFG and IGT) ‡
	Fasting glucose, mmol/l	5.4 (5.1, 5.8)	5.5 (5.1, 6.0)	<0.001	5.6 (5.2, 6.1)	<0.001
	Fasting insulin, pmol/l	7.4 (4.9, 10.8) 10.8 (7.3, 15.9) <0.001	8.5 (5.7, 12.6)	0.042
	2 hour post-glucose load glucose,	5.0 (4.2, 5.9)	5.5 (4.6, 6.6)	<0.001	5.8 (5.0, 7.1)	<0.001
	mmol/l
	2 hour post-glucose load insulin,	19.8 (11.7,	40.8 (23.6,	<0.001	24.3 (16.8, 41.0) <0.001
	pmol/l	34.5)	74.5)
	HOMA insulin resistance	1.8 (1.1, 2.7)	2.6 (1.7, 3.9)	<0.001	2.1 (1.3, 3.1)	0.003
	Total cholesterol	5.8 (5.1, 6.6)	5.7 (5.0, 6.4)	<0.001	5.4 (4.7, 6.2)	<0.001
	Fasting triglycerides, mmol/l	1.4 (1.0, 2.0)	1.6 (1.1,2.3)	<0.001	1.0 (0.8, 1.5)	<0.001
	HDL cholesterol, mmol/l	1.3 (1.0, 1.5)	1.2 (1.0,1.4)	<0.001	1.4 (1.2, 1.7)	<0.001
	Microalbuminuria, n/ total number	64/1055 (6%)	44/763 (6%)	0.79	20/288 (7%)	0.59
	with overnight urine collection †(%)
	Urine AER	4.0 (2.9, 6.1)	3.5 (2.5, 6.2)	<0.001	5.4 (3.6, 8.9)	<0.001
	Smoking status, % never/ ex/ current	29/40/32	74/11/15	<0.001	54/19/26	<0.001
	Physical activity score MJ/week	3.9 (1.2, 5.1)	3.5 (1, 4)	<0.001	3.7 (1.2, 5.8)	0.010

17 Figure : Southall and Brent studies mortality follow-up to 2007 a) Age-adjusted univariate and multivariate Cox regression analysis of associations of measured risk factors with CHD mortality in Europeans and Indian Asians.

		p †=0.5	p †=0.03	p †=0.04		p †=0.07	p †=0.02	p †=0.02
		6
		5
	Hazard ratio	2 3 4
		1
		0
		Fasting glucose 2 hour glucose 2 hour glucose,	Diabetes (new	Diabetes (new	Diabetes (new)
				multivariate		or known)	or known),
							multivariate
			Europeans	Indian Asians

b) Age-

adjusted univariate and multivariate Cox regression analysis of associations of measured risk factors with stroke mortality in Europeans, Indian Asians and African Caribbeans.

  Multivariate models adjusted for triglyceride, waist circumference, insulin, cholesterol, blood pressure and smoking p †= interaction p value: Indian Asians vs Europeans p ‡= interaction p value: African Caribbeans vs Europeans Diabetes defined according to World Health Organisation criteria, 1999[START_REF]Definition, Diagnosis and Classification of Diabetes Mellitus and its Complications. Part 1: Diagnosis and Classification of Diabetes Mellitus[END_REF]

			p †=0.04	p †=0.04	p †=0.04	p †=0.14	p †=0.07	p †=0.27
		20	p ‡=0.03	p ‡=0.02	p ‡=0.02	p ‡=0.04	p ‡=0.03	p ‡=0.12
		15
	Hazard ratio	10
		5
		0
			Fasting glucose 2 hour glucose 2 hour glucose,	Diabetes (new	Diabetes (new	Diabetes (new)
					multivariate	or known)	or known),
							multivariate
			Europeans	Indian Asians	African Caribbeans

√ √ √ √ 2 hour post-glucose challenge(if not known to have diabetes) Glucose, insulin Total cholesterol, triglycerides, HDL cholesterol √ √ √ -

Funding

The main SABRE study is supported by a joint programme grant from the Wellcome Trust and British Heart Foundation (BHF). The SABRE Cognitive function substudy is funded by the Wellcome Trust and the SABRE 3-D Heart substudy is funded by the BHF. The pilot follow-up study was supported by the BHF. The Southall study was supported by the UK Medical Research Council, the British Diabetic Association (now Diabetes UK), the Wellcome Trust and the BHF. The Brent study was supported by the UK Medical Research Council. *The SABRE Study group: Nish Chaturvedi (Imperial College London) (Principal Investigator) Norman Beauchamp (University of Washington, Seattle) Emma Coady (Imperial College London) Rory Collins (University of Oxford) Nita Forouhi (Medical Research Council Epidemiology Unit, Cambridge) Wladyslaw Gedroyc (Imperial College London) Ian Godsland (Imperial College London) Andrew Hattersley (Peninsula Medical School, University of Exeter) Alun Hughes (Imperial College London) Farrukh Majeed (Imperial College London) Jamil Mayet (Imperial College London) Paul McKeigue (University of Edinburgh) Naveed Sattar (University of Glasgow) Dean Shibata (University of Washington, Seattle) Therese Tillin (Imperial College London) Peter Whincup (St George's, University of London) Andrew Wright (Imperial College London)