Chapter 1

The Schrödinger equation 1.1 The history of quantum mechanics in small bits Niels Bohr, one of the father of quantum mechanics, claimed that 'if quantum mechanics doesn't profoundly shocked you, you haven't understood it yet", and in the same time, he confessed that he never understood it completely. There are numerous books devoted to the history and main concepts of quantum mechanics [53,42,30,24,45,11,41,32]. In such a history it is very difficult to define the starting point. In the particular case of the atom history, coming back to the Democritus ideas (four centuries before Christ) is, perhaps, a bit excessive!

The first relevant concepts were the Newton ideas about the light that he supposed composed by particles. However at the same period Huygens claimed that in fact light is a wave. This last theory was widely accepted after the experiments of Thomas Young (at the beginning of the nineteen century) and Augustin Fresnel a little bit later. This theory was definitively consolidated from the Maxwell works on electromagnetism.

The idea of the atom was firstly addressed by Robert Boyle (17 th century), by Newton in his works on physics and optics, by Lavoisier (18 th century) in his studies on combustion, by Dalton (at the beginning of the 19 th century) and by Gay-Lussac and Avogadro in the 19 th century.

Atoms were considered in physics by Maxwell and Boltzmann, in their theory of gases, the basis of statistical mechanics. But these ideas were not accepted by the scientific community, and some recognized scientist attacked this theory, not confirmed at that time experimentally, where the issue of reversibility was the major protagonist, motivating, apparently, Boltzmann suicide in 1906. The reality of the atoms was established in one of the three capital works published by Einstein in 1905. One of them concerned the relativity theory, the second one the photoelectric effect (from which he received the Nobel Prize in 1921) and the third one concerned the description of the Brownian motion based on the existence of atoms in continuous movement.

Chinesta, Francisco; Abisset-Chavanne Emmanuelle

At the end of the 19 th century there was a certain controversy about the essence of the radiation produced by a metallic filament driving an electrical current in a vacuum environment. J.J. Thompson in the Cavendish laboratory (founded by Maxwell in Cambridge, U.K.) conducted some key experiments leading to the electron discovery, that was the first elementary particle to be identified. For this discovery, J.J. Thompson received the Nobel Prize. Thompson, in the purest British tradition imagined the atom like a pudding or even better, like a watermelon where the negative electrons were uniformly distributed into a positive matrix. Research was very active at that time in some topics related to the nature of the X-rays or the radioactivity, with some of the most salient scientists involved in it: Henry Becquerel, Marie and Pierre Curie or Ernest Rutherford. The last one identified the alpha and beta radiations (the gamma one was discovered later) composed by positive and negative particles moving at high speed. The gamma rays seemed similar to the X-rays but showing lower wavelengths.

At the beginning of the 20 th century Rutherford (working at the University of Manchester) conducted some key experiments evidencing the alpha beam scattering when it impacted gold foils. To explain the observed results, incompatible with the Thompson atom model, Rutherford proposed a model for the atom consisting of electrons rotating around a densely positive nucleus, and he received in turn the Nobel Prize. However the Rutherford model had an important weakness because an accelerated charged particle emits radiation. The related energy loss should motivate its collapse, in contrast with the experimental evidence on the atoms stability.

At that time, another important research topic concerned the radiation coming from a black body. Classical models predicted an infinite energy. The main contribution in this field was provided by Planck who, using the statistical theory of thermodynamics developed by Boltzmann, concluded that the energy is quantified and related to the frequency ν from the expression: E = hν where h is known as the Planck constant and its value in the metric system of units is h = 6.626×10 -34 (J • s).

Planck works were considered in the model elaborated by Einstein (that deserved his Nobel Prize) concerning the photoelectric effect, that was explained from the introduction of the photon (light quanta). Thus, some centuries later, the Newton's idea about the discrete essence of the light (composed of particles) was renewed.

A new proposal of the atom model was given by Niels Bohr who suggested that electrons are located in orbitals having a permitted energy according to the energy quantification. To move from one orbit to another it must receive or loss an amount of energy corresponding to an integer number of quanta. This simple model allowed to describe successfully the chemical structure of the matter.

However, if we accept that light has a double character: particles and waves, why not extend this assertion to any other particle? This was the idea of Louis de Broglie who, starting from the Planck and Einstein results, established in his PhD thesis that momentum, p, and frequency, ν, are related from pc = hν, where c represents the speed of light. This relation has deep consequences, because in its left-hand side appears the momentum (characteristic of particles) while the right-hand side involves frequency (characteristic of waves), stating a subtle duality between particles and waves.

Obviously its experimental corroboration was not simple because in order to produce wave diffraction one needs to proceed with a diaphragm whose dimension must be of the same order than the wavelength λ . From the de Broglie's model it results that λ p = h. Thus, in order to obtain large enough wavelengths, momentum p must be small enough, as it is the case when considering lightweight particles. De Broglie during his PhD defense, answering a question of a sceptic member of his evaluation committee, suggested that his hypothesis could be verified by diffracting electrons in a crystal, scenario that makes compatible the wavelength (atomic distance) and the particles momentum.

The crucial experiment suggested by de Broglie during his PhD dissertation was finally conducted simultaneously by Davisson (in USA) and George Thompson (J.J. Thompson's son) in U.K. Both of them received the Nobel Prize. Thus, ironically, J.J. Thompson received the Nobel Prize for proving that electrons are particles and his son, George Thompson, received the same prize but now for proving that electrons are "also" waves!

It is important to note that interference, observed in these experiments, and easily understood within the wave framework, can be also explained from the Feynman path-integral, where each particle follows 'simultaneously" any possible trajectory while interacting with itself.

Another eminent and key scientist was W. Pauli, who introduced the fourth quantum number (the one related to the spin) as well as the exclusion principle which states that it is impossible to find two electrons in the same quantum state, and allowed a better understanding of the Bohr's atom model. At that time, two statistics were introduced, the one of Bose-Einstein (that applies on particles with integer spin that are not concerned by the exclusion principle) and the one due to Fermi-Dirac (that applies on particles with semi-integer spin and for which the exclusion principle applies). These statistics are of capital importance for describing the structure of matter as well as for explaining some exotic behaviors as the ones related to superconductivity and superfluidity.

The end of our brief overview concerns three attempts to describe the quantum mechanics reality. The first approach was introduced in Gottingen by the team led by Max Born, who, in collaboration with Heisenberg, proposed the matrix mechanics to describe quantum dynamics. The second approach was initiated in Cambridge by Paul Dirac who, starting from Heisenberg's works proposed a new algebra (the quantum algebra) which incorporates the matrix mechanics as a particular case. Finally, Erwin Schrödinger, inspired by de Broglie's works, proposed the introduction of a wavefunction describing the distribution of the particles in the whole space as well as the partial differential equation governing its evolution, known as waveequation. The last involves a continuous and unbounded medium (only the energy is discrete), where the evolution of the scalar and complex unknown field (the wavefunction) is governed by a partial differential equation, at first glance so simple! Even if this last approach can be considered within the general formalism of the quantum algebra, it seems conceptually simpler and more natural because of its connections with the well established physics of waves.

Obviously the three conceptual schemes, the first two more particle-oriented and the last one clearly wave-oriented, are equivalent as proved by Paul Dirac. However, today the Schrödinger formalism continues to be more popular, being preferred by a number of scientists working in the field of solid state physics and computational chemistry. Thus, this approach constitutes a powerful tool for describing the structure and mechanics of matter.

The wave-particle duality, the Heisenberg uncertainty principle (stating that a particle cannot "have" simultaneously a position and a momentum) and the reduction of the wavefunction (typically during an observation that produces the 'instantaneous" reduction of the wavefunction in favor of an apparent material-particle), among many others, are at the origin of a number of paradoxes (some of them still open) that continue to passionate scientists and non scientists! The interested reader can refer to some of the numerous available books previously referred.

Planck versus the ultraviolet catastrophe

In this section we revisit the classical theory of black-body radiation (Rayleigh-Jeans theory) and the so-called ultraviolet catastrophe. In order to solve this major issue, Planck introduced the energy quantization, considered later by Einstein to explain the photoelectric effect.

Rayleigh-Jeans theory

A stationary electric field in 1D writes

E = E 0 sin 2π x λ sin (2πνt) , (1.1)
where E 0 is the amplitude and λ and ν respectively the wavelength and the frequency, with c = λ ν (c being the speed of light). If the stationary wave is constrained to exist in an 1D cavity of length L with perfectly conductive boundaries, implying E(x = 0) = E(x = L) = 0, the following relation must be verified

2L λ = n, n = 1, 2, 3, • • • , (1.2) or ν = cn 2L , n = 1, 2, 3, • • • . (1.3)
Thus, the number of allowed frequencies within the interval [ν, ν + dν] becomes 2L c dν, in fact the double 4L c dν because we have two different polarizations. When considering a cubic cavity [0, L] 3 , and considering for a given direction expressed by the angles (α, β , γ) with respect to the cartesian axes

   λ = λ x cos α λ = λ y cos β λ = λ z cos γ , (1.4) the compatibility conditions read      2L λ x = n x 2L λ y = n y 2L λ z = n z , (1.5) or    2L λ cos α = n x 2L λ cos β = n y 2L λ cos γ = n z . (1.6)
Summing the square of all them (with cos 2 α + cos 2 β + cos 2 γ = 1) it results

2L

λ 2 = n 2 x + n 2 y + n 2 z → 2L λ = n 2 x + n 2 y + n 2 z , (1.7) or with r = n 2 x + n 2 y + n 2 z to r = 2L c ν. (1.8)
With the volume related to r being the one of the spherical shell, 4πr 2 dr divided by 8 for considering a single octant, and using the relation between r and the frequency ν, it results that the number of allowed frequencies in

[ν, ν + dν] is π 2 2L c 3 ν 2 dν,
(1.9) in fact the double because of the two possible polarizations.

The energy density ρ(ν) (the total energy divided by the cavity volume L 3 ) can be computed using the energy equipartition theorem, which associates a kinetic energy (E c ) of K b T /2 per degree of freedom:

ρ(ν)dν = 8πν 2 K b T c 3 dν, (1.10) whose integral ∞ 0 ρ(ν)dν = ∞ 0 8πν 2 K b T c 3 dν (1.11) diverges, i.e. ∞ 0 ρ(ν)dν = ∞, (1.12)
that constitutes the so-called ultraviolet catastrophe.

Planck theory

To avoid the divergence of the integral, Planck proposed a discrete energy distribution

E(ν) = nhν, n = 0, 1, 2, 3, • • • , (1.13)
which, using the Boltzmann theory, provides at equilibrium the following probability distribution:

p(n) = e -En K b T ∞ ∑ i=0 e - E i K b T , (1.14)
that is the probability, at equilibrium, to have n photons related to the frequency ν. Now, the energy average E ν reads

E ν = ∞ ∑ n=0 E n p(n) = ∞ ∑ n=0 nhνe -En K b T ∞ ∑ n=0 e -En K b T , (1.15) that using the change of variable x = e -hν K b T , becomes E ν = hν ∞ ∑ n=0 nx n ∞ ∑ n=0 x n = hν x + 2x 2 + 3x 3 + • • • 1 + x + x 2 + • • • = hνx 1 + 2x + 3x 2 + • • • 1 + x + x 2 + • • • . (1.16)
Taking into account the relations

1 1-x = 1 + x + x 2 + • • • 1 (1-x) 2 = 1 + 2x + 3x 2 + • • • , (1.17)
the energy average reads

E ν = hνx 1 -x = hν e hν K b T -1 . (1.18)
The energy density then reads

ρ(ν)dν = 8πν 2 E ν c 3 dν = 8πν 2 c 3 hν e hν K b T -1 dν, (1.19) whose integral does not diverge anymore ρ = ∞ 0 ρ(ν)dν = 8π 5 15h K b T h 4 , (1.20)
from which the Stefan-Boltzmann law ρ ∝ T 4 can be noticed.

An intuitive approach to the Schrödinger equation

Even if the Schrödinger equation could be assumed as a first principe (avoiding the necessity of deriving it), in this section we prefer deriving it following a nonrigorous procedure, but with the merit of being almost intuitive.

We consider as starting point some key results:

• The de Broglie equation establishing the relation between momentum p and wavelength λ : λ p = h; • The Einstein equation establishing the relation between energy and frequency (according to Planck): E = hν; • The general expression of a sinusoidal traveling wave:

Ψ (x,t) = sin 2π x λ -νt ; (1.21)
• The non-relativistic expression of energy:

E = p 2 2m +V, (1.22)
with V the potential. Because the fact of using the non-relativistic expression of the energy, the resulting Schrödinger equation becomes non-relativistic. Dirac considered its relativistic counterpart and derived the relativistic counterpart of the Schrödinger equation (the so-called Dirac equation).

Now, we assume a linear physics and focus on the simplest scenario that concerns a particle evolving in the space with constant potential V 0 . In absence of interactions (forces results from the potential gradient, here vanishing) the momentum and the energy should remain constant, according to the de Broglie and Einstein expressions pλ = h and E = hν. Thus, the energy expression (1.22) becomes

h 2 2mλ 2 +V 0 = νh, (1.23)
or by introducing k = 2π λ , ω = 2πν and h = h 2π , the energy expression (1.23) can be rewritten as h2

k 2 2m +V 0 = ω h, (1.24)
and the sinusoidal wave (1.21) reads

Ψ (x,t) = sin (kx -ωt) . (1.25)
Now, in absence of interactions, the evolution of the wavefunction Ψ (x,t) is expected to be given by a travelling wave like (1.25). Its space and time derivatives read

     ∂Ψ ∂ x = k cos(kx -ωt) ∂ 2 Ψ ∂ x 2 = -k 2 sin(kx -ωt) ∂Ψ ∂t = -ω cos(kx -ωt) , (1.26)
whose comparison with the energy expression (1.24) suggest

α ∂ 2 Ψ ∂ x 2 +VΨ = β ∂Ψ ∂t . (1.27)
To compute the expression of α and β , the wavefunction (1.25) is substituted into (1.27) with V = V 0 leading to

-αk 2 sin(kx -ωt) + sin(kx -ωt)V 0 = -β ω cos(kx -ωt),
(1.28) that has no solution for any choice of α and β .

To increase our chances we consider a more general form of the traveling wave

Ψ (x,t) = cos(kx -ωt) + γ sin(kx -ωt). (1.29)
Introducing this expression into (1.27) yields

(-αk 2 +V 0 + β ωγ) cos(kx -ωt) + (-αk 2 γ +V 0 γ -β ω) sin(kx -ωt) = 0, (1.30) that implies -αk 2 +V 0 = -β ωγ -αk 2 +V 0 = β ω γ .
(1.31)

By subtracting both equations (1.31) it results

-γ - 1 γ = 0, (1.32) that implies γ = - 1 γ , (1.33) that is, γ 2 = -1, or γ = ±i. Substituting this value in the first equation in (1.31) results -αk 2 +V 0 = ±iβ ω, (1.34)
whose comparison with the expression of the energy (1.24) leads to

α = - h2 2m β = ±ih . (1.35)
Without loss of generality, we choose β to be positive. We can now write the Schrödinger equation:

- h2 2m ∂ 2 Ψ ∂ x 2 +VΨ = ih ∂Ψ ∂t . (1.36)
It is important to note that as the Heisenberg uncertainty principle, discussed in the next chapter, establishes a huge difference between classical and quantum particles (the last cannot have a defined position and momentum simultaneously), in the wave description just described, the wavefunction seems to be also a quite strange wave, with a part of it defined in the complex dimension.

The Feynman approach

Feynman considered that for a free particle the wavefunction evolves from its initial state Ψ (y,t = 0) to the final one Ψ (x,t)

Ψ (y,t = 0) → Ψ (x,t).
(1.37)

The assumption of linearity implies that given two evolutions

Ψ I (y,t = 0) → Ψ I (x,t) Ψ II (y,t = 0) → Ψ II (x,t) , (1.38)
it results αΨ I (y,t = 0) + βΨ II (y,t = 0) → αΨ I (x,t) + βΨ II (x,t).

(1.39)

From the Schwartz kernel theorem we can write

Ψ (x,t) = y G (x, y;t)Ψ (y,t = 0)dy, (1.40)
with the kernel function G (•) independent on the origin of the coordinate system, i.e. G (x, y;t) = G (xy;t), and G (x, y;t) = G (xy) 2 ;t in the case of assuming isotropy. A dimensional analysis allows writing

G (x, y;t) = G (x -y) 2 m ht . (1.41)
Now, we consider a final state attained from two different paths

Ψ (z;t + t ) = y G (z, y;t + t )Ψ (y,t = 0)dy, (1.42)
and Using the previous results (1.41) and (1.44) with the normality condition

Ψ (z;t + t ) = x G (z, x;t )Ψ (x,t)dx, (
x ψ(x,t)Ψ (x,t)dx = 1, (1.45)
with Ψ the conjugate of Ψ , the expression of the kernel can be derived

G (x, y;t) = i t m 2π h e i t (x-y) 2 m h , (1.46)
where reversibility is ensured because i t = -i -t . Thus, if the path y → x implies a phase change

ϕ = (x -y) 2 m ht , (1.47)
x → z increases the phase change

ϕ = ϕ + (z -x) 2 m ht , (1.48)
but because the integral is performed at the intermediate state according to Eq. (1.44) an interference appears. Thus, if for George Tompson the electron interference experience was the proof of its wave nature, for Feynman the interference was simply the signature of a particle that is taking simultaneously all the possible trajectories interacting with itself (interaction is created by the integral in (1.44)). Now, from the kernel identified, the differential equation governing the evolution of the wavefunction can be obtained, and as expected it coincides with the Schrödinger equation.

The Schrödinger equation

In classical mechanics the evolution of a particle can be determined from the knowledge of its position r and velocity v (or its momentum p = mv, with m the particle mass) at a point of its trajectory. However, in quantum mechanics position and momentum can not be associated simultaneously to a particle, according to the Heisenberg uncertainty principle. Thus, more than speaking about position or momentum we should speak about probability distribution related to position P r = |Ψ | 2 and momentum P p = |Ψ p | 2 . The necessity of considering the square of Ψ or Ψ p results from the fact that both functions are complex and the resulting probabilities P r and P p must be necessarily real numbers.

Precisely because of the uncertainty principle, these two wavefunctions Ψ and Ψ p are not independent, the first one being the Fourier's transform of the second one. The properties of the Fourier transform guarantee that the higher the localization in space, the lower its localization in the momentum variable and vice-versa [51]. Of course, from the physical point of view this fact induces a conceptual difficulty because in classical mechanics two informations are needed to determine the evolution of the system (i.e. position and velocity) and the quantum system seems requiring only one Ψ (or Ψ p , because both are related from the Fourier transform). This paradox is only apparent because both wavefunctions are complex, and then they have a real and an imaginary part. Thus, the knowledge of a quantum system requires knowing the evolution of the wavefunction Ψ (or Ψ p ) introduced by Erwin Schrödinger in the twenties of the past century, that only needs the solution of the equation governing its distribution or its evolution in the transient case.

For the sake of simplicity we are first introducing the Schrödinger equation considering neither the relativistic effects nor the spin. If we assume a system composed of N P particles, the evolution of the joint wavefunction Ψ = Ψ (r 1 , r 2 , . . . , r N P ,t) is governed by the Schrödinger equation whose dimensionless form results [22] i ∂Ψ ∂t = -

N p ∑ p=1 1 2m p ∇ 2 p Ψ + N p ∑ p=1 N p ∑ k=p+1 V pk Ψ , (1.49)
where each particle p is distributed in the whole physical space r p = (x p , y p , z p ) ∈ R 3 and i= √ -1. The differential operator ∇ 2 p is defined in the conformational space of each particle, i.e.

∇ 2 p = ∂ 2 ∂ x 2 p + ∂ 2 ∂ y 2 p + ∂ 2 ∂ z 2 p .
(1.50)

The Coulomb's potential describing the inter-particle interactions writes

V pk = q p • q k ||r p -r k || , 1 Th (1.51)
where the masses m p are unity for electrons, charges q j are -1 for electrons and +Z j (atomic numbers) for nuclei. Equation (1.49) can be rewritten in the compact form

(H -T )Ψ = 0, (1.52)
where H denotes the Hamiltonian operator and T = -i ∂ ∂t . This equation was proposed, not deduced, and it is today considered as a first principle, like the Newton equation in classical mechanics.

It is important to note that Eq. (1.49) can be integrated from an initial condition Ψ (x,t = 0), leading to Ψ (x,t), with t ∈ (0, T ]. If we consider 0 < τ < T , and we do not observe the state of the system at t = τ, then for t > τ, Ψ (x,t) corresponds to the solution of Eq. (1.52) with the initial condition Ψ (x,t = 0). However, if we observe the system at time t = τ, the observation process modifies irreversibly the wavefunction, that is, Ψ (x, τε) = Ψ (x, τ + ε) (with ε → 0). Then, after the observation the system is described by the solution of Eq. (1.52) with the initial condition Ψ (x, τ + ε) that differs from the solution Ψ (x,t) computed from Ψ (x,t = 0).

By assuming the separation of variables the wavefunction can be decomposed according to

Ψ = Ψ r r 1 , . . . , r N p •Ψ t (t).
(1.53)

Introducing this expression into the Schrödinger equation (1.52) and dividing by

Ψ r •Ψ t yields Ψ t • H (Ψ r ) Ψ r •Ψ t - Ψ r • T (Ψ t ) Ψ r •Ψ t = 0, (1.54) or H (Ψ r ) Ψ r - T (Ψ t ) Ψ t = 0.
(1.55)

The first term depends on the space variables and the second one on the time, implying that both terms are equal to a constant E (that as discussed later represents the energy), that is

     T (Ψ t ) Ψ t = E H (Ψ r ) Ψ r = E . (1.56)
The integration of the first relation in (1.56) implies

Ψ t (t) = A e -iEt , (1.57)
and the second one leads to the following eigenproblem H (Ψ r (r 1 , . . . , r N p )) = EΨ r (r 1 , . . . , r N p ).

(1.58)

The eigenfunctions Ψ

(n) r (r 1 , . . . , r N p ) related to the eigenvalues E n define an orthogonal basis. Obviously, each function

Ψ (n) = A n e -iE n t •Ψ (n) r (r 1 , . . . , r N p ),
(1.59) is time-independent because the physical meaning of the solution is done by the complex norm of the wavefunction, that is

|Ψ (n) | = Ψ (n) •Ψ (n) = A 2 n e -iE n t • e iE n t •Ψ (n) r •Ψ (n) r = A 2 n Ψ (n) r •Ψ (n) r , (1.60)
where a denotes the conjugate of a.

Of course, the general transient solution can be written as

Ψ (r 1 , . . . , r N p ,t) = ∞ ∑ n=1 A n e -iE n t •Ψ (n) r (r 1 , . . . , r N p ), (1.61)
that obviously is not time-independent anymore.

In what follows, we assume a system composed of N e electrons and N n nuclei, and a time-independent solution of the Schrödinger equation. The Born-Oppenheimer model writes the total wavefunction of the combined system (electrons and nuclei) as a sum of products of wavefunctions related to the electrons and to the nuclei. Born and Oppenheimer showed that the total wavefunction can be reasonably approximated from a single product of a nuclei wavefunction Ψ n = Ψ n (R 1 , . . . , R N n ) and an electronic wavefunction Ψ e = Ψ e (r 1 , . . . , r N e ; R 1 , . . . , R N n ) that depends parametrically on the nuclei coordinates. Introducing Ψ = Ψ n •Ψ e into the Schrödinger's eigenproblem H (Ψ ) = EΨ it results

Ψ n • H e (Ψ e ) +Ψ e • H n (Ψ n ) = E Ψ e •Ψ n , (1.62)
or H e (Ψ e )

Ψ e + H n (Ψ n ) Ψ n = E, (1.63)
where the first term depends on the electron coordinates and the second one on the nuclei coordinates.

Ψ n = E -E e , (1.66) that results in (H n + E e )Ψ n = E Ψ n .
(1.67)

The difference between the dynamics of electron and nuclei is derived in the following way: (i) the nuclei are typically tens of thousands times heavier that the electrons; (ii) the particles constituting the molecules are in equilibrium so, in average, they have similar kinetic energies (equipartition theorem); (iii) thus the ratio of the square of their velocities (electrons and nuclei) will be roughly the inverse ratio of their masses (velocity ratio of the order of hundred); (iv) then the wavelength associated with the nuclei is a hundred of times lower that the one corresponding to the electrons. The last conclusion comes from the fact that according to de Broglie λ -1 n ∼ p n = m n v n ≈ 10 4 m e 10 -2 v e = 10 2 p e ∼ 10 2 λ -1 e (where • n and • e refer respectively to the nuclei and electrons properties).

Thus, electrons see stationary nuclei and the nuclei see electrons distributed (in a sort of mean). From now on we assume the nuclei fixed in the physical space and write

H (Ψ (r 1 , . . . , r N e ; R 1 , . . . , R N n )) = E Ψ (r 1 , . . . , r N e ; R 1 , . . . , R N n ), (1.68)
where the parametrical dependence of the electronic distribution on the nuclei position has been emphasized. The Hamiltonian can be written as

H (Ψ ) = - N e ∑ e=1 1 2 ∇ 2 e Ψ + N e ∑ e=1 N n ∑ n=1 V en Ψ + N e ∑ e=1 N e ∑ e =e+1
V ee Ψ , (1.69)

where

V ee = 1 ||r e -r e || , (1.70)
and

V en = - Z ||r e -R n ||
.

(1.71)

Relations between position and momentum wavefunctions

As previously indicated Ψ (x,t) and Ψ p (p,t) are Fourier transforms of each other in order to fulfill the Heisenberg uncertainty principle addressed later. Thus, proceeding in the 1D case for the sake of simplicity we have

Ψ p (p,t) = 1 √ 2π h ∞ -∞ Ψ (x,t)e -i px h dx = F (Ψ (x,t)), (1.72)
where F (•) refers the Fourier transform and the use of the constant h (h = h 2π ) is introduced into the Fourier transform definition for addressing later the Heisenberg uncertainty principle. The inverse transformation writes

Ψ (x,t) = 1 √ 2π h ∞ -∞ Ψ p (p,t)e i px h dx. (1.73)
This definition of the Fourier transform is not practical when using standard convolution expressions, but this is not an issue here and moreover the convolution expression can be modified accordingly.

Using the Fourier transform properties we can write

p n Ψ p = F h i n ∂ n Ψ ∂ x n . (1.74)
In what follows we will also make use of the Parseval's theorem that for

Ψ p = F (Ψ ) and Φ p = F (Φ) states ∞ -∞ Ψ p • Φ p d p = ∞ -∞ Ψ • Φ dx.
(1.75)

Calculating expectations

As soon as the electronic distribution Ψ is known one could compute the one related to the electronic momentum Ψ p and from both the different expected values of position, momentum, energy, etc. Imagine a simple system consisting of a single electron in the one-dimensional space, i.e. r = x. Thus, writing: P x = |Ψ | 2 and P p = |Ψ p | 2 the expected value for the position reads

x = R xP x dx = R x Ψ (x) •Ψ (x) dx, (1.76) with R = (-∞, ∞).
The analogue expression for the momentum results

p = R pP p dx = R p Ψ p (p) •Ψ p (p) d p,
(1.77) that using Eq. (1.74) and the Parseval theorem results

p = R pP p d p = R Ψ (x) h i ∂Ψ (x) ∂ x dx.
(1.78)

Analogously, we can compute p 2

p 2 = R Ψ (x) h i 2 ∂ 2 Ψ (x) ∂ x 2 dx. (1.79)
Thus, for a polynomial g(p) it results

g = R Ψ (x) g h i ∂ ∂ x Ψ (x) dx, (1.80)
where

g h i ∂ ∂ x refers the replacement of p in the polynomial g(p) by the differential operator h i ∂ ∂ x .
By considering f (x) and g(p) and using the linearity it results

f + g = R Ψ (x) f + g h i ∂ ∂ x Ψ (x) dx.
(1.81)

When applied to the Hamiltonian H (x, p)

H = p 2 2m +V (x), (1.82) it results H = R Ψ (x)H Ψ (x) dx = R Ψ (x) V (x) - h2 2m ∂ 2 ∂ x 2 Ψ (x) dx.
(1.83)

All the previous results can be easily extended to 3D scenarios. Thus, as soon as the wavefunction Ψ (x,t) is known, the different expectations related to position, momentum and/or energy can be easily computed.

Heisenberg uncertainty principle

If we assume for a while (and without loss of generality) x = 0 and p = 0,

(∆ x) 2 = (x -x ) 2 = x 2 ,
(1.84) and similarly

(∆ p) 2 = p 2 .
(1.85)

If we define the integral (1.86) 1.7 Heisenberg uncertainty principle its expansion reads

I (α) = R α h ∂ ∂ x -x Ψ 2 dx ≥ 0,
I (α) = R α h ∂ ∂ x -x Ψ • α h ∂ ∂ x -x Ψ dx = α 2 h2 R ∂Ψ ∂ x ∂Ψ ∂ x dx + R x 2 ΨΨ dx -α h R ∂Ψ ∂ x xΨ dx + R xψ ∂Ψ ∂ x . (1.87)
Now, by integrating by parts the first term in the right-hand side and taking into account that the wavefunction vanishes when x → ∞, it results h2

R ∂Ψ ∂ x ∂Ψ ∂ x dx = -h 2 R Ψ ∂ 2 Ψ ∂ x 2 dx = (∆ p) 2 .
(1.88)

On the other hand, integrating by parts the third term yields

R ∂Ψ ∂ x xΨ dx = - R Ψ Ψ + x ∂Ψ ∂ x dx = -1 - R Ψ x ∂Ψ ∂ x dx, (1.89)
leading to the polynomial expression

I (α) = α 2 (∆ p) 2 + α h + (∆ x) 2 ≥ 0. (1.90)
Since this quantity is nonnegative, the discriminant of the quadratic polynomial

h2 -4(∆ p) 2 (∆ x) 2 ≤ 0 (1.91)
is negative, that leads to the usual expression of the Heisenberg uncertainty principle

(∆ p) 2 (∆ x) 2 ≥ h2 4 .
(1.92)

When x = 0 and/or p = 0 we obtain the same result as proved in [51]. All the previous results can be easily extended to 3D scenarios.

The main consequences are that quantum particles are quite different to their classical counterpart, because in quantum theory positon and velocity cannot exist simultaneous, if position is perfectly defined (it exists in a classical sense) the 'quantum particle" has a totally undetermined velocity and vice-versa.

It exists another uncertainty relationship relating energy and time (even if the last is not an operator). This expression gives an approximate value for the length of time taken for the energy to change by one standard deviation. In order to get an accurate energy measurement it should change slowly. This result is of major interest for explaining the quantum tunnel effect.

Observable and its time evolution

The expected value of any observable F (Hermitian operator on a complex vector space, like position, energy, ...),

F = R Ψ FΨ dx, (1.93)
can be written in a more compact form as

F = (Ψ , FΨ ), (1.94)
where

(Ψ , Φ) = R Ψ • Φ dx. (1.95)
When considering the time-independent Hamiltonian observable H = (Ψ , H Ψ ), the time evolution of its expectation is given by

d H dt = ∂Ψ ∂t , H Ψ + Ψ , H ∂Ψ ∂t , (1.96)
where the time-independent nature of H implies ∂ H ∂t = 0. By integrating by parts and taking into account both the Hermitian character of observables (the Hamiltonian in the present case) and the fact that Ψ vanishes when

x → ∞ gives d H dt = ∂Ψ ∂t , H Ψ + H Ψ , ∂Ψ ∂t , (1.97)
O = R Ψ OΨ dx, (1.99)
or using the notation previously introduced O = (Ψ , OΨ ).

(1.100)

The time evolution of observable O reads

d O dt = ∂Ψ ∂t , OΨ + Ψ , O ∂Ψ ∂t , (1.101)
that using again the Schrödinger equation gives

d O dt = - i h H Ψ , OΨ + Ψ , - i h OH Ψ , (1.102)
that can be rewritten as

d O dt = i h ((H Ψ , OΨ ) -(Ψ , OH Ψ )) , (1.103) or d O dt = i h ((Ψ , H OΨ ) -(Ψ , OH Ψ )) , (1.104)
that finally leads to

d O dt = i h (Ψ , (H O -OH )Ψ ) , (1.105)
or its compact counterpart

d O dt = i h [H , O] , (1.106)
where the so-called commutator [•] (Poisson bracket) is defined as

[H , O] = H O -OH . (1.107)
When the observable depends explicitly on time it results that

d O dt = dO dt + i h [H , O] .
(1.108)

The Ehrenfest theorem

From these results, it is very easy to prove the Ehrenfest's theorem that states

F = d p dt p = m d x dt , (1.109)
establishing a perfect analogy between classical and quantum mechanics, in the sense that quantum expectation fulfills classical Newton mechanics.

To prove the previous relations we consider both of them

d p dt = i h p 2 2m +V (x), p , (1.110) and d x dt = i h p 2 2m +V (x), x , (1.111)
respectively, where the fact that position and momentum operators do not have time dependences was taken into account. By developing the first relation, taking into account that the momentum commutes with itself and substituting the momentum by p → -ih ∂ ∂ x , it results that

d p dt = R ΨV (x) ∂Ψ ∂ x dx - R Ψ ∂ ∂ x (V (x)Ψ ) dx. (1.112)
Developing the second integral in the previous equation, it finally results

d p dt = - R Ψ ∂V (x) ∂ x Ψ dx, (1.113) or d p dt = - ∂V (x) ∂ x = F , (1.114)
that generalizes Newton's second law. Now, coming back to the second expression in Eq. (1.109) and by using the commutation relation (that again reflects the Heisenberg uncertainty principle)

[p, x] = -ih, (1.115) it results d x dt = i h p 2 2m +V (x), x = i h p 2 2m , x = i 2mh p [p, x] + [p, x]p = 1 m p . (1.116)
Again, all the previous results can be easily extended to the 3D case.

1.9 Charge density and interatomic potentials: The Hellmann-Feynman theorem

As soon as the electronic distribution is known, the electron density ρ e (r) can be easily computed by applying

ρ e (r) = R 3(Ne-1) Ψ •Ψ dr 1 • • • dr e-1 dr e+1 • • • dr N e , (1.117)
allowing us to compute the electronic density at each point of the space by adding all the electron contributions according to

ρ(r) = N e ∑ e=1
ρ e (r).

(1.118)

Now, the Hellmann-Feynman theorem [51] leads to a direct interpretation of the electronic effects on the nuclei in terms of the classical electrostatics. The applied force on a nucleus k results

f k = -∇ k ( V nk + V ek ), (1.119)
where V nk is the electrostatic contribution inter-nuclei

V nk = N n ∑ n=1,n =k Z 2 R n -R k , (1.120)
and V ek is the one associated with the electronic distribution:

V ek = -Z   R 3 ρ(r) r -R k dr   .
(1.121) Thus, it seems clear that knowing the electronic distribution, the electronic density can be computed and from it the effects that the electronic distribution has on the nuclei. The solution of the Schrödinger equation allows computing the exact interatomic potentials that could be then used to perform accurate molecular dynamics simulations. However, the multi-dimensional character of the Schrödinger equation makes very difficult such solution even for moderate populations of electrons. Two possibilities exist: (i) the one that consists in introducing some simplifications in the treatment of that equation; or (ii) the one related to the proposal of empirical (or quantum inspired) inter-atomic potentials to be used in the molecular dynamics framework as discussed in Chapter 3. Ab initio simulations are performed using quantum-based potential whereas molecular dynamics uses empirical potentials.

Before analyzing the state of the art on the computational treatment of the Schrödinger equation we are introducing a last key concept, the one related to the Pauli exclusion principle.

The Pauli exclusion principle

The wavefunction of many-electrons must reflect the fact that electrons are indistinguishable. Electrons do not know that we have labeled them. If we use r 1 to describe the coordinates of one electron, and r 2 for another, then

|Ψ (r 1 , r 2 , r 3 , . . . , r Ne )| 2 = |Ψ (r 2 , r 1 , r 3 , . . . , r N e )| 2 , (1.122)
that is, the probability distribution of electrons cannot depend on the way that we have labeled them. Thus, if Π is any permutation of the N e electronic coordinates, then Π |Ψ (r 1 , r 2 , r 3 , . . . where the exclusion Pauli principe states that for electrons a = -1. Thus, the most general statement of the Pauli principle for electrons establishes that an acceptable wavefunction for many electrons must be antisymmetric with respect to the exchange of the coordinates of any two electrons.

If we imagine a system composed of two non-interacting electrons (V e 1 e 2 = 0) the resulting wavefunction can be written in the one-term separated form

Ψ (r 1 , r 2 ) = Ψ 1 (r 1 ) •Ψ 2 (r 2 ),
(1.127) but because of the Pauli exclusion constraint this function must be adjusted to ensure its antisymmetry (a = -1 in Eq. (1.126)) by considering the Slater determinant

Ψ (r 1 , r 2 ) = Ψ 1 (r 1 ) •Ψ 2 (r 2 ) -Ψ 1 (r 2 ) •Ψ 2 (r 1 ) = Ψ 1 (r 1 ) Ψ 2 (r 1 ) Ψ 1 (r 2 ) Ψ 2 (r 2 ) , (1.128)
where the use of the determinant ensures that changing the electron's label corresponds to exchanging the two associated columns in the determinant, and the associated change in the wavefunction sign. However, this strategy introduces some technical difficulties by increasing the number of terms involved in the wavefunction expression.

Until now we have not included anything in the description to take into account the electron spin. One could think that as there is no mention of the spin in the Hamiltonian, the spin should have little relevance to the energy of the system. However, the Pauli principle applies to the exchange of all the coordinates (space and spin) so in order to comply with it the spin must be introduced into the particle coordinates.

If we do not introduce the spin into the description of a system composed of two electrons, the solution must verify: Ψ (r 1 , r 2 ) = -Ψ (r 2 , r 1 ), but if the spin is considered, the expression Ψ (r 1 , r 2 ) = Ψ (r 2 , r 1 ) holds if the two electrons have opposite spin.

We can give a simple solution to this issue by including the spin into the particles coordinates by defining the extended coordinates x i from

x i = (r i , s i ) = (x i , y i , z i , s i ), (1.129)
where s i is the spin coordinate of electron i, that can take two values that we denote by α and β (in fact -1/2 and 1/2 in the case of electrons). Thus, two electrons can be allocated in the same orbital as soon as they have different spin. Pauli principle is of major importance for explaining the electronic structure of matter and also atomic bonding.

Remark. When one is solving the transient Schrödinger equation, if the initial condition verifies the Pauli exclusion principle, the computed transient solution will verify the antisymmetry without considering any particular treatment. However, when the time-independent solution is searched the antisymmetry must be enforced, and the most common way to enforce it is the use of the Slater's determinant. It is important to note that the computation of transient solutions needs very efficient integration schemes able to ensure the energy conservation while guaranteeing the stability.

On the numerical solution of the Schrödinger equation

Due to the high dimensionality of the Schrödinger equation its solution is only possible for very reduced populations of electrons. For this reason different approximated methods have been proposed and extensively used. We are describing in the next chapter the two most widely used: the Hartree-Fock (HF) method and the Density Functional Theory (DFT), the first one can be applied only for a moderate number of electrons and it has been extensively used in the context of quantum chemistry to analyze the structure and behavior of molecules. However, in the case of crystals, the number of electrons becomes too large to make possible their simulation by the direct solution of the Schrödinger equation or by using the Hartree-Fock technique. In this context the most successful technique was and continues to be the DFT. Both techniques (HF and DFT) are un practice approximated techniques that work in some cases and in other cases could become crude approximations.

The three techniques (direct Schrödinger solution, HF and DFT) can be applied for solving both the time-independent and the transient Schrödinger equations. Transient solutions are very common in the context of quantum gas dynamics (physics of plasma) but they are more scarce in material science when the structure and properties of molecules or crystals are concerned. For this reason, in that follows we are focusing on the solution of the time-independent Schrödinger equation which leads to the solution of its associated multidimensional eigenproblem.

Before considering in the next chapter the two approximated techniques (HF and DFT) we consider here the most natural and accurate one, the one consisting in the direct solution of the Schrödinger equation. In the time-independent case, usually only the so-called ground distribution (the one related to the minimum energy) is searched. In any case, the high dimensional spaces in which the Schrödinger equation is defined leads to the curse of dimensionality. Nowadays it is widely accepted that classical discretization methods, based on the use of a grid (or a mesh), fail when the space dimension approaches few tens. Some attempts to address multi-dimensional equations have been proposed: the simplest choice consists in using stochastic techniques (described in Chapter 3.2) because in this case the computation complexity does not scale exponentially with the space dimension. However, these simulations are expensive and introduce statistical noise. Moreover, only the moments of the distribution can be computed accurately, because an accurate description of the distribution itself requires too many trajectories of the stochastic process. A second possibility consists in using sparse grid techniques but nowadays the use of these techniques only allows solving models in spaces rarely exceeding few tens. A third possibility consists in employing separated representations (at the origin of the so-called Proper Generalized Decomposition) and that allowed an accurate and fast solution of models involving many particles [6]. Even if the use of separated representations in computational mechanics is quite unusual, its use has been extensively considered in the context of quantum chemistry, in particular it is at the basis of the Hartree-Fock method described in the next chapter. Within the context of separated representations the main issue when addressing the solution of the Schrödinger equation is related to the antisymmetry of the wavefunction enforced by using the Slater determinant whose number of terms explodes with the space dimension (number of particles involved in the quantum system).

In order to illustrate the limits of classical discretization techniques for addressing the direct solution of the Schrödinger equation we consider two quotations extracted from [35]:

"For the solution of the time-dependent problems, different levels of approximation have been used, which range between the direct discretization of the TDSE -time dependent Schrödinger equation -as the most precise but computationally most expensive choice, and time-dependent density functional theory (TDDFT), which is appealing from a computational point of view but turned out to be too crude an approximation to capture important features of the problem, ..." "The present state-of-the-art of numerically solving the time-dependent Schrödinger equation directly for realistic laser pulses is limited to two electron systems. The most successful calculations involve the largest massively parallel computers available. It is clear that the direct solution of the linear time-dependent Schrödinger equation has reached its computational limits" or the analysis addressed in [38]: ' ... the theory of Everything is not even remotely a theory of every thing. We know this equation [the Schrödinger equation] is correct because it has been solved accurately for small number of particles -isolated atoms and small molecules -and found to agree in minute detail with experiments. However it cannot be solved accurately when the number of particles exceeds about 10. No computer existing, or that will ever exist, can break this barrier because it is a catastrophe of dimension. If the amount of computer memory required to represent the quantum wavefunction of one particle is N then the amount required to represent the wavefunction of k particles is N k . It is possible to perform approximate calculations ... but the schemes for approximating are not first-principles deductions but rather art keyed to experiment ... "

One could think that this pessimistic perspective is based on the fact that the complexity scaling with N k is due to the fact of using a grid instead of using separated representations. In some cases, the use of a separated representation circumvents the curse of dimensionality, leading to a complexity scaling with N • k instead of N k , however some quantum systems explore the whole configurational space whose dimension corresponds to N k (for example the spin glass problem considered in [10] consisting of a set of k quantum spins interacting by random Heisenberg exchanges where the configuration space dimension becomes 2 k and obviously the system can explore the whole configuration space). In these cases no solution exists! Thus, we must assume the existence of numerically tractable and intractable quantum systems. In what follows we are considering only the first ones and addressing their solutions by using approximated techniques.

  1.43) that imply the composition rule G (z, y;t + t ) = x G (z, x;t )G (x, y;t)dx, (1.44) illustrated in Fig. 1.1.

Fig. 1 . 1

 11 Fig. 1.1 Evolution from Ψ (y;t = 0) to Ψ (z;t + t ).

  , r Ne )| 2 = |Ψ (r 1 , r 2 , r 3 , . . . , r N e )| 2 , (1.123) that implies just two possibilities ΠΨ (r 1 , r 2 , r 3 , . . . , r Ne ) = Ψ (r 1 , r 2 , r 3 , . . . , r N e ), (1.124) or ΠΨ (r 1 , r 2 , r 3 , . . . , r Ne ) = -Ψ (r 1 , r 2 , r 3 , . . . , r N e ),(1.125)that can be written as ΠΨ (r 1 , r 2 , r 3 , . . . , r Ne ) = aΨ (r 1 , r 2 , r 3 , . . . , r N e ), (1.126)


Laurence Bernard

Muriel Bonnet

Carole Delavaud

Mylène Delosière

Anne Ferlay

Hélène Fougere

Benoit Graulet

Advances in nutritional strategies to alter the lipid fraction of ruminant milk for a healthier life

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

Advances in

Milk lipids (3 to 5%) are present in the form of small droplets called milk fat globules emulsified in the aqueous phase (87%). This lipid fraction contains a core of nonpolar lipids (mainly triglycerides ; 98% ; source of energy) coated into the milk lipid globule membrane (MLGM) composed of polar lipids mainly phospholipids and minor specific proteins, and is considered as a source of bioactive molecules. Lipid-soluble vitamins (A as retinol and β-carotenes, D, E, K; ~ 40-60 µg/g lipids) and xanthophylls must also be considered as high value micronutrients of the lipid fraction. Among fatty acids (FA) carried by triglycerides and phospholipids, some may exert negative effects (12:0, 14:0, and 16:0) when consumed in excess, whilst others have potentially beneficial effects (anteiso-15:0, cis-9-18:1, and 18:33) on human health or putative positive effects as shown in rodent models (cis-9,trans-11 CLA). Over the last decade, numerous nutritional experiments have been carried out to enhance the milk fat content of a number of specific FA recognized as having beneficial effects on human health and to decrease those that may have detrimental effects. Moreover, the milk vitamin E and A (and β-carotene) content is affected by the nutritional state and amount and type of forage in ruminants. Lastly still little is known on effect of nutrition on the composition of lipid, and even less on protein, of the MFGM. Because nutrition is the main lever of the milk fat plasticity, the aim of this review is to report the recent data on the effect of diets on the milk lipid fraction of ruminants. We will also update data on the impact of diets on plasma lipid composition and mammary lipogenic and desaturation pathways that determine the milk fat composition. We will focus on recent data using new techniques of lipidomics and proteomics to identify the major and minor components of the lipid fraction and spectral methods to predict FA composition. Finally, we will pay particular attention to differences in the responses to certain nutritional factors of the milk fat content and composition in dairy ruminants, which offers an original model of study to decipher the underlying mechanisms.

  Nutritional Strategies to Alter the Lipid Fraction of Ruminant Milk for a Healthier Life L. Bernard, M. Bonnet, C. Delavaud, A. Ferlay, H. Fougère, B. Graulet, UMR1213 Herbivores, INRA, VetAgro Sup, Clermont université, université de Lyon Saint-Genès-Champanelle, France (12 pt, Centered) Milk and dairy products are recognized as important sources of nutrients in human diets.


Jaap J Van Milgen

Marie-Hélène Pinard-Van Der Laan

Isabelle Schwartz-Cornil

Yuksel Cagla

Valérie Kaya

Heuzé

M Cerf

Multidisciplinary approaches to livestock production

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

The emphasis on interdisciplinary research is far from new: promoting it was at the core of the DGRST French research plan in the 70's and of the Hawkesbury University experience in Australia. It resulted in the emergence of the INRA SAD division in the 80's. Today, many funding agencies ask for interdisciplinary work teams. Well recognized journals such as Science give room to interdisciplinary results. Doctoral schools value interdisciplinary training even if this remains controversial regarding recruitment standards. More epistemological reflexion is available to address the foundations of interdisciplinary scientific work. Anyway, practising interdisciplinary work and managing interdisciplinary projects remain a big challenge. There is no clear-cut recommendations or on-the shelf tools which can be transferred to newcomers. Many researchers who joined interdisciplinary projects find it difficult, disappointing. They acknowledge that the project is more a collection of disciplines working on separate tasks with loose coordination in terms of renewed understanding of the issue addressed in the project. Why is it so? I will rest on my experience at the interface between agronomic and social sciences to address the dynamic of interdisciplinary work in practice. I will put emphasis on various dynamic patterns and stress the need for intermediary objects, shared learning and reflexivity along the interdisciplinary journey. I will illustrate that fruitful interdisciplinary work takes place at the border of the disciplines but often questions core assumptions underlying its main stream. Therefore, taking part to interdisciplinary work put the participants in a risky position in their own community as the legitimacy of their work becomes discussed. To escape this uncomfortable position, researchers often trigger a debate on the balance between excellence and relevance of the scientific work. I will then conclude by focusing on the management of interdisciplinary projects. I will argue that people involved in the management of interdisciplinary work act as brokers. As for any brokering work this implies to have the ability and the curiosity to navigate between different points of view on a reality and to support the cross fertilization between these points of view.

Multidisciplinary approaches to livestock production J. Van Milgen 1 , M.H. Pinard-Van Der Laan 2 , E. Schwartz 3 , Ç. Kaya 4 and V. Heuzé 5 1 INRA-Agrocampus Ouest, UMR Pegase, Le Clos, 35042 Rennes, France, 2 INRA, UMR GABI, Domaine de Vilvert, 78352 Jouy-enJosas, France, 3 INRA, UR VIM, Domaine de Vilvert, 78352 Jouy-enJosas, France, 4 EFFAB, Dreijenlaan 2, 6703 HA Wageningen, the Netherlands, 5 Association Française de Zootechnie,16,rue Claude Bernard,75231 Paris,France;jaap.vanmilgen@inra.fr Livestock production is an essential component of a sustainable food supply. The dimensions of sustainability combined with the complexity of biology call for multidisciplinary approaches to assess the functioning of (components of) livestock production systems. Horizon 2020 projects such as Saphir and Feed-a-Gene are funded by the European Commission and have a multidisciplinary approach towards livestock production (e.g. nutrition and genetics, genetics and health, novel management strategies and socio-economic aspects). The objective of this session is to address and discuss the challenges and opportunities in multidisciplinary research in livestock production. As there is no 'one-size-fits-all' solution, how can different actors and stakeholders make collectively best use of (disciplinary) knowledge and levers to make livestock production more sustainable?


Florentin Millour

Damien Ségransan

Jean-Philippe Berger

Gilles Duvert

Fabien Malbet

Interferometry Basics in Practice: Exercises

Keywords: Optical long baseline interferometry, visibility, phase, U V coverage, VLTI, ASPRO

The following exercises aim to learn the link between the object intensity distribution and the corresponding visibility curves of a long-baseline ptical interferometer. They are also intended to show the additional constraints on observability that an interferometer has. This practical session is meant to be carried out with the ASPRO software, from the Jean-Marie Mariotti Center, but can also be done using other observation preparation software, such as viscalc from ESO.

There are two main parts with series of exercises and the exercises corrections. The first one aims at understanding the visibility and its properties by practicing with simple examples, and the second one is about U V coverage.

Notes: -for the binary model, the Flux Ratio is F secondary / F primary , and Rho & Theta are the angular separation (") and position angle (degrees) of the binary.

-R.A. and Dec are usually set to zero while Flux is set to one.

-"0" means you have to fill the parameter value a with zero. Be careful to put a 0 instead of leaving it blank, otherwise the software will crash ! Plotting a uniform disk visibility curve: Given a star with a 2 milli-arcsecond (mas) photospheric radius, use the model function to plot the visibility versus the projected baseline length (baseline radius). For this purpose, you can use the MODEL/FIT.UV Plots & Source Modeling menu.

Zero visibility:

At what baseline does the visibility become equal to zero (you can refer to Berger & Segransan, 2007, for example)? Use this number to evaluate the disk diameter.

Diameter uniqueness: Can you measure a unique diameter if your visibility is non-zero but you know the star looks like a uniform disk? If yes, how?

Exercise 2: Binary star.

When the amount of data you get is a low number of visibilities, the object's complexity for your interpretation cannot be too high. Therefore, binary models are often used to understand the data when asymmetries happen to be proved by means of interferometry (by a non-zero closure phase) or by indirect clues (a polarization of the target, for example). Therefore, one has to understand the behavior of such a model.

Plotting a binary star visibility curve: Display the visibility and phase as a function of projected baseline (using the file strip-K-60) of a binary with unresolved components with 4 mas separation, a flux ratio of 1, and a position angle of 30 degrees. Do the same thing with different separations. Comment on the result.

Varying the flux ratio: Using the previous model, now vary the flux ratio from 1 to 1e-6. Comment on how the dynamic range requirement to detect the companion translates into visibility and phase constraints?

Phase versus visibility: Would the phase alone be sufficient to constrain the binary parameters?

Exercise 3: Circumstellar disk.

The last model we will see in this practice session is a Gaussian disk that can, in a first approximation, simulate a circumstellar disk, or an optically thick stellar wind. The idea here is to understand how visibilities change with source elongation.

Plotting a Gaussian disk visibility curve: Display the visibility curve of a disk which is assumed to have an elliptical Gaussian shape (model E GAUSS).

Use the minor and major axes (parameters 4 & 5) to simulate an inclination. The display should be done for several 30,45,60,90)).

Aspherity and visibility variations: Comment on how the aspherity induced by the inclination changes the visibility function at a given projected angle.

Exercise 4: Model confusion and accuracy.

If the baseline you have chosen is too long or too small relative to the typical size of your source, this may cause problems when you try to interpret your data. Here you will see why.

Plotting several model visibilities: Use the model function to compute the visibility of a star with a uniform disk brightness distribution (2 mas radius), circular Gaussian disk (1.2 mas radius), and binary (flux ratio 1, 1 mas separation, 45 • PA) with the baseline stripe strip-K-60. No superposition of the plot is possible, so use the show plot in browser option, save it as a postscript file, and compare the different files afterwards. Compare their visibilities at 100 m in the K band.

Model confusion at small baselines: How can we distinguish between these various models? What about measurements at 200 m? What do you conclude?

The role of measurement accuracy: Does the measurement accuracy play a role in such model discrimination?

Which baseline for which purpose: Construct a 2-component model in which a central, unresolved star (POINT) is surrounded by an inclined, extended structure. You can use an elliptical Gaussian distribution (E GAUSS) for this purpose (minor and major axes in the range 0.5 to 15 mas).

Try two scenarios:

• an extended source easily resolvable but with a flux contribution much smaller than the star; • a smaller extended source but with a larger flux contribution.

What are the best baseline lengths for estimating the size and relative flux contributions with an interferometer?

Exercise 5: Choosing the right baselines.

Given a specific object's shape, one can determine how a baseline constrains a given model parameter. We will see this aspect here. In order to determine the parts of the UV plane which constrain the model most, one can make use of the first derivative of the visibility with respect to a given parameter (e.g. derivative of visibility versus diameter). Gaussian disk: Do the same exercise using a Gaussian disk.

Exercise 6: An unknown astrophysical object.

The wavelength at which an object is observed is also important. This exercise attempts to illustrate this point.

Loading and displaying a home-made model: Load the fits table fudisk-N.fits corresponding to the simulation of a certain type of astrophysical object (here, a disk around an FU Orionis object) using OTHER/Display a GDF or FITS image menu. If the color scale is not appropriate, check the Optional parameters button and select another color scale. Notice what the contrast of the object (angular units in radians) is.

Computing the visibilities of a home-made model: Compute the visibility of the model in the N-Band with MODEL/ FIT.UV Plots & Source Modeling/USE HOMEMADE MODEL. To do so, select the appropriate grid strip-N-60 in the Input Information menu. Use UV EXPLORE to plot the visibility amplitude versus the spatial frequency radius.

Comparing visibilities for different wavelengths: Repeat these operations in the K band, then the H and J ones. Compare the visibility profiles. Conclude on the optimal wavelength to observe the object with the VLTI (maximum baseline is 130 m today).

Exercise 7: Play with spectral variations, closure phases, etc.

Bonus exercise: Try to guess what this model shows just by looking at the visibilities (please do not cheat!).

Plotting visibilities and closure phase versus wavelength: Using a given model of a binary star (γ 2 Vel, file gammaVelModelForAspro.fits), try to plot visibility and closure phases as a function of wavelength (see the OTHER ... Export UV table as OI fits and OTHER ... OI fits file explorer menus).

Qualitative understanding: Compare the obtained visibilities with what you would get with an ASPRO model of one Gaussian and one uniform disk of diameters 0.5 mas, a separation of 3.65 mas, and an angle of 75 degrees. What do you conclude?

Looking at the solution: After your conclusions, you can look at the model by opening it with a fits viewer (for example, fv1 ).

From model to visibility (Correction)

The author of this paper carried out the previous exercises using ASPRO (and an image processing software for superposition of the graphs: GIMP) to give an idea of what one should get with ASPRO when following the previous exercises. Please try to do the exercises yourself before reading these corrections.

Exercise 1: The diameter of a star. The place where one can infer a unique diameter from a single visibility measurement for a uniform disk is where the measured contrast is above 0.15. Indeed, below 0.15, several diameters can be inferred from this single visibility measurement (as shown by the red line).

Zero visibility:

The visibility for a uniform disk of diameter a is given by the following expression:

V (ρ) = 2 J 1 (πaρ) πaρ (1)
ρ = B/λ (B and λ in meters) being the spatial frequency, a being the star diameter (in radians), and J 1 the 1 st order Bessel function. Therefore, the value for which the visibility becomes zero is B = 1.22λ/a.

Here, the visibility zeroes around the 280 m baseline. This gives an approximately 1.89 mas diameter for the star, close to the 2 mas input.

Diameter uniqueness: The right plot in Fig. 1 gives a hint of the answer: There are parts of the visibility function which are monotonic (above the red line). In these parts, one visibility gives a unique solution to the diameter of the star. In the parts below the red line, a given visibility corresponds to many different solutions (as the line crosses several points of the curve). Therefore, there is a lower limit on the visibility value (V 0.15) where one can infer a unique diameter from a unique measurement.

Exercise 2: Binary.

Plotting a binary star visibility curve: The visibility and phase functions of a binary star are periodic since the image is made of Dirac functions. One can see what can be expected for a 4 mas-separation binary star in Fig. 2. One can see that the visibility does not have a cosine shape, but has sharp changes at visibility 0 for a 1 to 1 binary (black line).

Varying the flux ratio : For other flux ratios (0.8 in red, 0.5 in blue, and 0.1 in green), both the phase and visibility get smoother, and the contrast of the variations gets dimmer. Please note that these are visibility plots made with "AMP" and not "AMP 2" in ASPRO. Phase versus visibility: To see how visibility or phase can constrain a binary star model, one can just try to change the baseline orientation and see how visibility and phase vary. In Fig. 3, bottom-right, one can see the result of such exercise. One has seen that the phase is sensitive to the contrast of the binary (Fig. 2), as is the visibility, but it is also sensitive to the position angle (both of the binary star and of the baseline) and the binary separation. Therefore, the phase can be used instead of the visibility to constrain the binary parameters! Right: Visibility versus base length for the minor (black, upper curve) and major (blue, lower curve) axes of the Gaussian disk. Please note that the visibility is lower in the direction of the major axis and greater in the minor axis direction.

Exercise 3: Circumstellar disk.

Plotting a Gaussian disk visibility curve: The example of this exercise is a disk of 2x4 mas and a position angle of 60 degrees. The resulting UV map and cuts in the UV plane are shown in Fig. 4.

Aspherity and visibility variations:

As one can see, for a given baseline and a varying position angle, the visibility goes up and down, the minimum corresponding to the major axis and the maximum corresponding to the minor axis. One has to note (and can check) that the phase function for such a model is zero.

Exercise 4: Model confusion and accuracy.

Plotting several model visibilities: In Fig. 5 different plots are superimposed using the GIMP software. The uniform disk is in black, the Gaussian in blue, and the binary in red.

Model confusion at small baselines: Two green lines represent the 100 m and 200 m baselines. One can see the importance of multi-measurements at different baselines to be able to disentangle the different objects' shapes. With only one visibility measurement, one will never be able to distinguish between If the accuracy is not better than about 0.01, then no distinction can be made between the different models. For the 200 m baseline (second green vertical line), an accuracy of 0.1 or better is sufficient to discriminate between the different models. Note, however, that one needs more than 1 point to be able to really distinguish between the models. these different models. With the two visibility measurements, one can see that it will be possible to disentangle the different models if both very different baselines AND a sufficient accuracy can be reached.

The role of measurement accuracy: Here, if one has visibility error bars of 0.05, one can distinguish between the sources using the 100 and 200 m baselines, but not with an accuracy of 0.3. This importance of accuracy in model confusion is illustrated by Fig. 6. The 1st plot (on the left) is a central point with 90% of the total flux contribution and a large (15mas) Gaussian disk around, accounting for 10% of the flux. Errors bars of 0.1 would prevent one from finding the Gaussian component, and the measured visibility (0.9 ± 0.1) would be compatible with a completely unresolved star. The right graph shows the same but with the flux ratio inverted (90% of the flux to the disk and 10% to the central star). In this case, the visibilities would be compatible with a fully resolved, extended component, given an accuracy of 0.1 on the visibility, and the point source (the central star) would not be detected. So, not only the number of baselines but also the accuracy of the measurement is of high importance to distinguish between different models.

Which baseline for which purpose: Here, one can see that since there is an unresolved source in the object image (a central source), measuring the visibility at long baseline directly provides the unresolved flux /resolved flux flux ratio.

If one wants to get information on the disk itself (size, shape, etc.), the shorter baselines are more appropriate, since the visibility will vary according to the source shape. The idea here is to use the nice feature of ASPRO that is able to plot the derivatives of visibility versus the different parameters of the input model. To do so, one has to go to the UV explore panel, select U as X data and V as Y data, and finally select d(AMP )/d(4) in the Plot what... part. This will plot the derivative of the visibility amplitude versus the 4 th parameter (radius, as shown in table 2). Fig. 7 shows the model visibility and its derivative, relative to the model size, for a Gaussian disk and a uniform disk.

Uniform disk: The size used is 2 mas. The derivative peaks where the visibility slope versus baseline is the largest. This means that for a small baseline change, a large visibility change will be observed; i.e., the biggest constraint will be applied to the corresponding model in the model fitting process.

One can see that the optimal baseline is 100 m for the uniform disk. One should know that the optimal baseline to constrain a given model corresponds to a visibility of about 50%. Very important to know: baselines that are too short or too long will not reveal much information about the source size or shape.

Gaussian disk: The size used is the same as before: 2 mas. The optimal baseline is about 50 m for the Gaussian disk. Therefore, using different baselines, one will be able to both disentangle the model shape (Sharp -UD -or smooth -Gauss -edges?) and constrain the typical size (as seen before). Please note that, as for the previous disk model (Exercise 3), the disk is elongated, and therefore, the large visibilities will correspond to the minor axis of the model, whereas the low visibilities will correspond to the major axis.

Computing the visibilities of a home-made model: The lower-left part of Fig. 8 displays a UV map of the model shown in the upper part. This map is very similar to the one shown in Fig. 4, which indicates that the disk mostly looks like a Gaussian disk.

Comparing visibilities for different wavelengths: The lower-right part of Fig. 8 shows the visibility for the minor axis with different wavelengths: J (black, lower curve), K (blue, middle curve), and N (red, upper curve). This shows which wavelength to use to observe the object: the K-band allows both high visibilities for the extended component and low visibilities for the detailed structure of the disk, in the range of the VLTI offered baselines (16-130 m).

Exercise 7: Play with spectral variations, closure phases, etc. This is a bonus exercise. To manage it, one will need the newest local version of ASPRO to cope with closure phases (at the time of writing this correction, the web version of ASPRO did not have all functionalities necessary to do this exercise). Therefore, one is not obliged to reach this point.

Plotting visibilities and closure phase versus wavelength: In Fig. 9, one can see the visibilities for a given observational setup (A0-D0-H0) and the closure phase. The author has used, on purpose, an array with aligned baselines to illustrate this exercise.

Qualitative understanding: There are a number of indicative clues to qualitatively understand the shape of the observed object: • The different visibilities are decreasing with baseline. Therefore, the object is barely resolved by the interferometer. One cannot qualitatively distinguish between a uniform disk, a Gaussian disk, or a binary star, but one can say the object is resolved at the largest baseline ( ≈130m) and therefore has a size of about 2mas. • The non-zero closure phase gives information about the asymmetry of the object. Here, one has a non-zero but very small closure phase. The only simple model known from this practice session which gives a non-zero closure phase is a binary star model. The fact that the closure phase is not 180 degrees gives the additional information that the flux ratio is not 1/1.

Therefore, only qualitatively looking at these data one can say:

• The object is likely to be a binary star,

• the separation is about 2mas,

• the flux ratio is not 1/1,

• so far, one cannot say if each component has been resolved or if there is a third component.

Looking at the solution: Fig. 10 shows the image of the model used in this exercise. As one can see, the components are somewhat resolved, but probably not enough to be detected, and a third component is present. Therefore, the qualitative analysis is not enough, and one has to perform a quantitative analysis to characterize all of these components. 3 Observability and UV coverage What you will need for this particular practice session:

You will need to use ASPRO and the catalogs named sampleSources1.sou and sampleSources2.sou provided with this practice session (you can copy this information into text files, see below). To set up ASPRO, please refer to the 1st part of this practice session about visibilities and model fitting.

!----------------------------------------------------- ! sampleSources1.sou !---------------------------------------------------
!---------------------------------------------- ! sampleSources2.sou !---------------------------------------------- STAR_40
EQ 2000.000 16:00:0.0 40:00:0.0 STAR_30 EQ 2000.000 16:00:0.0 30:00:0.0 STAR_20 EQ 2000.000 16:00:0.0 20:00:0.0 STAR_10 EQ 2000.000 16:00:0.0 10:00:0.0 STAR_0 EQ 2000.000 16:00:0.0 0:00:0.0 STAR_-20 EQ 2000.000 16:00:0.0 -20:00:0.0 STAR_-40 EQ 2000.000 16:00:0.0 -40:00:0.0 STAR_-60 EQ 2000.000 16:00:0.0 -60:00:0.0 STAR_-80 EQ 2000.000 16:00:0.0 -80:00:0.0 You should first load the catalog named sampleSources2.sou. It contains 7 stars of R.A. 5:00:00 and of different declinations. In this section, you will make intensive use of the OBSERVABILITY/UV COVERAGE menu of ASPRO.

Exercise 3: UV tracks for a North-South baseline

We will now study the UV coverage of the sources with OBSERVABIL-ITY/COVERAGE, UV coverage & PSF. • Select a large 3-telescope array configuration (in the WHERE menu).

• Visualize the observability of the targets (including constraints on delay lines). Look at the OBSERVABILITY/COVERAGE, Observability limits due to delay lines panel to see why the observability range is smaller with 3 telescopes than with 2 telescopes.

UV tracks for different configurations: In the UV coverage panel, try to add several 3-telescope configuration. For that you need to un-check the reset frame button. As an example, you can select 4 configurations i.e. A0-G1-J6, G2-J1-D2 and A1-B2-C1.

Beam shape: You can then display the "dirty beam" (the same as in radioastronomy !) by using the Display PSF panel.

Exercise 6: Radius measurement of a star (uniform disk)

Here you will play with configurations and "real" observations. You will have a set of stars you want to observe. You must figure out if they are observable and choose the best observing setup to accurately measure the diameters.

In this part, you should load the catalog named sampleSources1.sou. Select an instrument and the K band (2.1µm). You should also select an observing period and an optimal array configuration to determine the radius of the targets with the highest accessible accuracy. In this section you will make intensive use of the WHAT & Object Model menu (or UV Model/FIT, Source modeling menu) and OBSERVABILITY/COVERAGE menu of ASPRO. Use the appropriate uniform disk model to either display the amplitude, the phase of the visibility, or the derivatives with respect to the diameter to visualize which part of the UV plane really constrains the model.

Optimizing the observability of a series of sources: Can you find a setup which fits well all the stars together? For that purpose, you must find a night and configuration which fits well all the stars characteristics for observability, visibility level, delay lines constraints, and UV tracks.

Radius measurement: Can one determine the radius of these stars?

More details about the object: Can one determine phenomena that occur at higher spatial frequencies, like limb darkening?

Knowing the limitations: What accuracy do you need to fulfill your objectives?

Exercise 7: Binary parameter determination

In fact a "mistake" was introduced in the previous list: the star HD 68273 is a binary star (real name γ 2 Velorum). First load/re-load the catalog named sampleSources1.sou and then select star HD 68273. Let us consider it as a binary system with the properties summarized in Table 4. • Select the baseline G2-G1

• Visualize the UV coverage and the amplitude. Does this baseline constrain the parameters of the binary? Plot the visibility as a function of time. • Select the baselines A0-M0. Visualize the amplitude, the phase, and their derivatives. • Does this baseline constrain the parameters of the binary? • Plot the visibility as a function of time.

• What do you notice about the baseline orientation / the binary system position angle?

4 Observability and UV coverage (Correction)

The same procedure as before was used to produce these corrections: ASPRO was used for getting the figures and GIMP to over-plot the graphs together for illustration. As in the previous section, please try the exercises first before reading these corrections.

Exercise 1: Setting up an observation

After having set the date, time, place, and target (Fig. 13), you can start using ASPRO and its many features to check and prepare observations. Fig. 14 is what you should get in your shell when setting View Object Catalog correctly. Exercise 2: Observability of sources at different declinations and delay line constraints

Which stars are observable? Fig. 15 shows the observability for the second catalog of sources (sampleSources2.sou). The gray area corresponds to the night, and the black lines correspond to the observability of sources by their height above the horizon (here 30 • ). First, you can see that a source at Aspro> !-----------------------------------------------------! sampleSources1.sou !------------------------------------------------- Exercise 3: UV tracks for a North-South baseline

The UV tracks for the targets of the 2nd catalog (sampleSources2.sou) are shown in Fig. 16. The North-South baseline gives roughly North-South UV tracks, but the limitations due to the delay line limits are quite severe here since a very long baseline (G1-J6) was used. This can be seen in the very short accessible UV tracks for the 20 and 30 • declination targets. One can see here that the 0 • declination target gives a line in the UV plane. Exercise 4: UV tracks for an East-West baseline UV tracks shape: For the answer to this question, please refer to Millour (2008). The UV tracks correspond to the projection of a circle (due to the Earth rotation) on an inclined plane (the plane of sky) and are therefore arcs of ellipses.

UV tracks and target declination: Fig. 17 now shows the East-West baseline. As one can see, East-West baselines will never give a North-South UV projection on the sky. Also, the total range of UV plane projected angles is maximum for high declination targets (either positive or negative). As for the previous case, a 0 • target gives a straight line in the UV plane and, therefore, provides less coverage than a high declination target.

Exercise 5: UV tracks for a 3-telescope-array

Observability: First, one needs to check the observability when taking a big triangle (B5-J6-M0, Fig. 18, left panel). As one can see, the observability constraints are much more stringent than before, using a lower number of telescopes. The Observability limits due to delay lines panel (right side) now gives meaningful information; i.e., splitting the constraint by delay line. Here, one can see that the most constraining baselines are the B5-J6 and M0-B5, and the M0-J6 baseline does not too much constrain the observability. UV tracks for different configurations: In Fig. 19, the UV tracks for the different triangles and different object declinations (0 • , -20 • and -40 • from top to bottom) are shown. One can see that many aspects will affect the UV coverage:

• declination -which affects the quantity of different position angles.

• delay lines constraints -which prevent one from observing with large baselines (bottom graph). • available baselines -which limits the available stations due to the interferometer possibilities.

Beam shape: One can see that the many aspects raised before will affect the UV coverage and, therefore, the quality of the PSF (plotted using the show dirty beam option):

• declination affects the quality of different position angles and gives irregular secondary lobes to the PSF. • The delay line constraints prevent one from observing with large baselines (bottom graph) and widen the PSF (loss of angular resolution). • The available baselines give an elongated PSF in the E-W direction for the VLTI, since the longest baseline is in the N-S direction.

Exercise 6: Radius measurement of a star (uniform disk)

Optimizing the observability of a series of sources: The first step is to set up an observation date in order to be able to observe all targets together in one night. The optimal date of observation would be the 15 th of December, but the moon is full on the 20 th . However, one can choose this date (Fig. 20) since interferometry is insensitive to the moon phase. I also chose the UT1-UT3-UT4 triplet since there is a faint target in my sample (NGC 1068) and checked that it does not put too many additional constraints on the observability of the source due to delay lines.

Radius measurement: First of all, you need to look at the standard accuracy of the instrument you will use to be able to know what you can expect from your observations. The ESO Call for Proposal3 gives you information about AMBER and MIDI (the two offered instruments) accuracy you can expect. For AMBER in P81, the accuracy is 3% for an uncalibrated visibility point; i.e., about 5% for a calibrated one. If you look at the different targets here (I have taken a diameter of 3 mas for NGC 1068, see Fig. A.2), you can see that with this accuracy, both the star Betelgeuse and HD 68273 (γ 2 Vel) are unreachable for diameter measurement. The star HD 81720 will be measurable but with a poor accuracy on the diameter. Therefore the targets you will really be able to observe using these baselines are Achernar, NGC 1068, and partially HD 81720. For the two other stars (HD 68273 and Betelgeuse), you will need other configurations (a short baseline triplet for Betelgeuse and the longest available baselines for HD 68273) to reach your goal of measuring a diameter.

More details about the object: You should also notice that limb-darkening measurements, which need at least one point in the second lobe of visibility, are not achievable, except for Betelgeuse. Finally, one baseline setup is not sufficient to measure all the star diameters: you will need at least a 3-baseline setup to reach your proposal goal (one short baseline setup for Betelgeuse and one very long baselines needed for HD 68273 and HD 81720 to reach a better accuracy).

Knowing the limitations: By browsing the ESO call for proposals4 , you can find that the current AMBER accuracy is 0.03 on the raw visibilities. This means 0.05 on calibrated measurements. Therefore, as seen in Fig.

A.2, only Achernar and NGC1068 will allow one to "easily" measure a non-zero or non-1 visibility. HD68273 and Betelgeuse will make the measurement very difficult, as the expected visibility is very close to 0 or 1. Then, HD81720, whose visibility at maximum baseline is 0.85, will be marginally resolved and only an upper limit to the diameter will be measurable.

Exercise 7: Binary parameter determination

If the reader has reached this point carried out all of the exercises without any problems, he is now an expert in long-baseline stellar interferometry and can do this exercise without any difficulty ;-)

  Uniform disk: Choose a uniform disk model. What is the most constraining part in the UV plane? For this exercise, in the UV EXPLORE panel, use V versus U, check the under-plot model image option, and choose the appropriate derivative in the plot what... line.

  Plotting a uniform disk visibility curve: The figure produced by ASPRO should look like Fig. 1, left.

Fig. 1 .

 1 Fig. 1. Left: The uniform disk visibility function cut from exercise 1. Right:The place where one can infer a unique diameter from a single visibility measurement for a uniform disk is where the measured contrast is above 0.15. Indeed, below 0.15, several diameters can be inferred from this single visibility measurement (as shown by the red line).

Fig. 2 .

 2 Fig. 2. Left:The visibility amplitude for a separation of 4 mas and different flux ratios (from the lower to the upper curves: 1 to 1 in black, 0.8 to 1 in red, 0.5 to 1 in blue, and 0.1 to 1 in green). Right: The visibility phase for the same separation and flux ratios.

Fig. 3 .

 3 Fig. 3. Top: U V map showing the previous binary star visibility modulus. This visibility shows characteristic modulation stripes perpendicular to the binary orientation. Bottom-Left: Visibility modulus as a function of base length for the different projected stripes, in different colors (the number of sine arches increases with the different orientations): red is 90 • relative to the binary orientation, green is 60 • , blue is 30 • and black is 0 • . Bottom-right: Visibility phase as a function of base length for the same stripes as before.

Fig. 4 .

 4 Fig. 4. Left: U V map showing the visibility amplitude for a 2x4 mas elongated Gaussian disk. The black and blue lines correspond to the right part of the figure.Right: Visibility versus base length for the minor (black, upper curve) and major (blue, lower curve) axes of the Gaussian disk. Please note that the visibility is lower in the direction of the major axis and greater in the minor axis direction.

Fig. 5 .

 5 Fig. 5. Illustration of model confusion for different baselines and models: blue (top curve) is a Gaussian disk, red (bottom curve) is a binary star, and black (middle curve) is a uniform disk. The first green vertical line represents a 100 m baseline.If the accuracy is not better than about 0.01, then no distinction can be made between the different models. For the 200 m baseline (second green vertical line), an accuracy of 0.1 or better is sufficient to discriminate between the different models. Note, however, that one needs more than 1 point to be able to really distinguish between the models.

Fig. 6 .

 6 Fig. 6. Left: Visibility cut for a point star accounting for 90% of the total flux and a 15 mas Gaussian disk accounting for 10% of the flux. Right: Visibility cut for a point star accounting for 10% of the total flux and a 15 mas Gaussian disk accounting for 90% of the flux.

Fig. 7 .

 7 Fig. 7. Top-left: Gaussian disk visibility map, as a function of U V coordinates. Top-right: Visibility derivative versus the model size (FWHM), showing where the visibility varies more with U V coordinates. Bottom-left: Visibility map, but using a uniform disk. Bottom-right: Visibility derivative versus model size (FWHM) for a uniform disk model.

Fig. 8 .

 8 Fig. 8. Top: Image of the disk model used in exercise 6 (shown using the fv tool) Bottom-Left: Visibility map showing the main elongation of the disk, perpendicular to the main elongation of the visibility function. Bottom-Right: Visibility of the disk for different wavelength regimes: red (top curve) is N-band, blue (middle curve) is K-band, and black (bottom curve) is J-band.

Fig. 10 .

 10 Fig. 10. Image of the binary+wind-wind collision zone model used in this exercise.

Exercise 1 :

 1 Setting up an observation Set the date: In the WHEN menu, Date & Time Setup, put the date 28-AUG-2007 and time 14:00:00. Set the place: In the WHERE menu, select VLT, 2 Telescopes. Set the target: In the WHAT menu, choose Use Object catalog and select the file sampleSources1.sou. If you use the web version of ASPRO, you need to have an account on the JMMC server and to copy your files beforehand using the File Management panel. Check the settings: Check with the WHAT menu View Object catalog and look at the result in the xterm window. Note: you can do this step only with a local version of ASPRO as the web version has no access to a terminal.Exercise 2: Observability of sources at different declinations and delay line constraints First, we will check the observability of the sources with OBSERVABIL-ITY/COVERAGE, Observability of Source. Set the minimum elevation to 30 • , check the Plot the twilight zones button, and use UT1 and UT2. When everything is done, press the GO button.Which stars are observable? Is the chosen date appropriate for observing all stars together? Delay lines limitation: Now, go to OBSERVABILITY/COVERAGE, Observability limits due to delay lines. How does it change the observability? Compare the observability with UT1-UT4 and G1-J6. What do you conclude?Sampling the UV plane with the VLTI This goal of this section is to see how the UV coverage changes with baseline orientation and source declination.

Fig. 11 .

 11 Fig. 11. The VLTI stations. Photo: Gerhard Hüdepohl • Select the star at declination -20 and set the wavelength to 2 microns. In the Telescopes & Stations panel, select a 2-telescope baseline oriented N-S (cf. Fig. 11 ) and have a look to the shape of UV coverage you get. • Change stars, going from positive to negative declination and see what happens (you can over-plot the graphs by unchecking the RESET FRAME button in the Telescopes & Stations).Hint: Look at the orientation of the Earth in Fig.12.

Fig. 13 .

 13 Fig. 13. How to set the date, place, and object. Top: the date setup window appearing when pressing the WHEN menu. After changing the date and time, just press GO. Middle: the WHERE menu, where one can choose between all the supported interferometers. Bottom: the WHAT menu. The Get CDS Object... is very convenient for quickly finding a target.

Fig. 14 .

 14 Fig. 14. Display of the catalog in the terminal window.declination +40 is not observable at all, due to the latitude of Paranal: about 30 • .

Fig. 15 .

 15 Fig. 15. Left: Observability of sources at different declinations for the UT1-UT4 baseline (roughly East-West baseline). Right: Observability of sources for the G1-J6 baseline (North-South).

Fig. 16 .

 16 Fig. 16. U V plane tracks due to the Earth rotation for a N-S baseline (G1-J6) and for different declinations: cyan for -20 • , blue for 0 • , green for 10 • , red for 20 • and black for 30 • .

Fig. 17 .

 17 Fig. 17. U V plane tracks due to the Earth rotation for an E-W baseline (A1-J2). The color-coding for the different declinations is the following: black for -80 • , cyan for -40 • , blue for -20 • , green for 0 • , red for 20 • and yellow for 40 • .

Fig. 18 .

 18 Fig. 18. Left: Observability of sources for a big triangle of baselines (B5-J6-M0). The constraints due to the delay lines (red lines) are now larger than for 2 telescopes. Right: Detail of the delay line constraints: the B5-J6 and M0-B5 are the most constraining baselines.

Fig. 19 .

 19 Fig.19. Top: Sketch of the stations used in this example. Middle-Left: U V tracks for a 3-telescope observation on a 0 • declination object and several telescopes configurations: A1-B2-C1, A0-G1-J6 and G2-J1-D2, corresponding roughly to 3 observing nights using AMBER and with complete freedom in the stations selection. Bottom-Left: The corresponding "dirty beam", or what would look like the PSF if one would observe a point-source with a telescope having an aperture similar to the previous U V track. Middle: Same as 1 st plot, but for a -20 • declination target. Bottom-Middle: The corresponding dirty beam. Middle-Right: Same as before but for a -40 • declination target. Note that the biggest triangle does not appear in this plot, as the constraints due to the delay lines prevents from observing with these telescopes. Bottom-Right: The corresponding dirty beam.

Fig. 20 .

 20 Fig. 20. Observability of the 1 st catalog.

Fig. 21 .

 21 Fig. 21. Top-Left: Achernar visibilities for the largest baselines, given a radius of 2.53 mas. Top-Right: Same for NGC 1068 and a diameter 3 mas. Middle-Left: Same for Betelgeuse, diameter of 44 mas. Middle-Right: Same for HD68273, diameter of 0.5 mas. Bottom-Left: same for HD81720, diameter of 0.93 mas.

Table 2

 2 Description of the parameters of the ASPRO model software module. Be sure to regard the parameter units.

	name / parameter 1 st	2 nd 3 rd	4 th	5 th	6 th
		(")	(") (no unit) (unit)	(unit)	(unit)
	POINT	R.A. Dec Flux	0	0	0
	C GAUSS	R.A. Dec Flux	Diameter (")	0	0
	E GAUSS	R.A. Dec Flux	Maj. diam. (") Min. diam. (") Pos. Ang. ( • )
	C DISK	R.A. Dec Flux	Diameter (")	0	0
	E DISK	R.A. Dec Flux	Maj. diam. (") Min. diam. (") Pos. Ang. ( • )
	RING	R.A. Dec Flux	In diam. (")	Out diam. (") 0
	U RING	R.A. Dec Flux	Diameter (")	0	0
	EXPO	R.A. Dec Flux	Diameter (")	0	0
	POWER-2	R.A. Dec Flux	Diameter (")	0	0
	POWER-3	R.A. Dec Flux	Diameter (")	0	0
	LD DISK	R.A. Dec Flux	Diameter (")	cu	cv
	BINARY	R.A. Dec Flux	Flux Ratio	Rho (")	Theta ( • )

http://heasarc.gsfc.nasa.gov/lheasoft/ftools/fv/

http://www.eso.org/sci/observing/proposals/

http://www.eso.org/sci/observing/proposals/

APPENDIX A Practical considerations

A.1 ASPRO rules of thumb ASPRO is an optical long-baseline stellar interferometry tool intended to help the observations preparation. One can find it on the website http://www.jmmc.fr. Note that all exercises can be done at home with an internet connection, given that ASPRO can be launched via a java web interface.

Launching ASPRO on the web: The ASPRO launch is set up in five simple steps:

(1) go to http://www.jmmc.fr Launching ASPRO on a local computer: If your computer is one of the rare ones to have ASPRO locally installed, then the launch is even simpler:

(1) In a command line, type aspro @oipt (2) In the menu Choose..., select the ASPRO version you want to use. Here, we will use the Full ASPRO interface version. Do not worry, restarting ASPRO and entering the different parameters again does not take very long!


Mathieu Besson
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J A M Van Arendonk

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Economic and environmental impacts of genetic improvement in fish depend on limiting factors

Keywords: selection, bioeconomic model, life cycle assessment, feed efficiency, thermal growth coefficient

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The target of fish breeding is to increase economic profit by producing faster growing fish with lower feed intake. Genetic improvement, therefore, is also expected to reduce environmental impacts of fish farming. However, little is known about the economic and environmental consequences of improving growth rate and feed efficiency in fish farming. At farm level, genetic improvement of a trait affects feeding strategy and/or management practices according to the limiting production factor. Evaluating the impact of a genetic improvement requires therefore, (1) to model the whole farm operations via a bioeconomic model and (2) to calculate the environmental impacts along the production chain using Life Cycle Assessment. We calculated environmental and economic values (ENV and EV) of thermal growth coefficient (TGC) and feed conversion ratio (FCR) in two different farming systems: recirculating aquaculture system (RAS) of catfish and sea cages (SC) of sea bass. In RAS, the limiting factors are nitrogen treatment capacity of the bio-filter and fish density. In SC, the limiting factors are fish production quota and oxygen availability. ENV and EV were calculated using the difference in profit and in environmental impacts per kg of fish produced between the current population mean for both traits (µ t ) and the next generation of selective breeding (µ t + Δ t ). The results show that ENV and EV are both dependent on limiting factors. Improving TGC is economically and environmentally beneficial in RAS and SC, only when the limiting factors are respectively rearing density (RAS) and oxygen availability (SC). There is no benefit in improving growth rate when nitrogen treatment capacity or production quota are the limiting factors. On the other hand, improving FCR always increases profit and decreases environmental impacts in RAS and SC irrespective of the limiting factor. These results emphasize the importance of calculating environmental and economic values in the right farming context in order to develop efficient future breeding programs in aquaculture. FCR is the most important trait increasing profitability and decreasing environmental impacts in fish farming. This result emphasizes, the need for further studies aiming at better characterising the genetic bases of feed efficiency.