Brandon L Findlay
email: brandon.findlay@concordia.ca

A geometric derivation of the electron, proton, and neutron magnetic moments

est destinée au dépôt

Introduction

The electron, proton, and neutron possess a number of physical properties, such as mass, magnetic moment, quantum spin, and (for the former two particles) charge [START_REF] Data Group | Review of Particle Physics[END_REF] . Within the Standard Model, all of these are considered axiomatic: they are not predicted by the theory but instead constrain it [START_REF] Burgess | The Standard Model: A Primer[END_REF] . To go beyond the Standard Model these properties must therefore be derived or demonstrated to be fundamental.

In the macroscopic realm, a moving charge creates a magnetic field according to µ 0 j = ∇ × B, while a charge loop creates a magnetic moment according to τ = µ × B 3 . While the magnetic moment of fundamental particles appears to stem from the movement of charges [START_REF] David | The Nature of Intrinsic Magnetic Dipole Moments: What Can the Famous 21 Cm Astrophysical Spectral Line of Atomic Hydrogen Tell Us about the Nature of Magnetic Dipoles? CERN Yellow Reports: Monographs[END_REF] , attempts to model the movement of classical current loops at the microscopic scale have proven incomplete, as this leads to angular acceleration and an accelerating charge emits light [START_REF] Griffiths | Introduction to Electrodynamics[END_REF][START_REF] Larmor | On the theory of the magnetic influence on spectra; and on the radiation from moving ions[END_REF][START_REF] Podgoršak | Rutherford-Bohr Model of the Atom[END_REF] .

However, there are at least two instances in which an accelerating charge does not emit light: the oscillation of complementary charges within a standing wave and the movement of a continuous charge within a circuit [START_REF] Larmor | On the theory of the magnetic influence on spectra; and on the radiation from moving ions[END_REF][START_REF] File | Observation of Persistent Current in a Superconducting Solenoid[END_REF][START_REF] Goedecke | Classically Radiationless Motions and Possible Implications for Quantum Theory[END_REF] . Of the two the former is more reasonable as a physical model, but given the wealth of theory behind the modeling of charged circuits in this work I will focus on the latter, bringing in elements of the former only when necessary to further the derivation.

The electron

The electron as a solenoid

Begin with an infinitely thin, ethereal wire with no electrical resistance, curved into a two loop solenoid whose path is described by,

x (θ) = r * sin θ cos θ y (θ) = r * sin θ sin θ z (θ) = r e * sin θ (1)
Where the radius of the loops is equal to r = λ e 4π and the distance between loops is set by the classical electron radius, r e = 2 * 2.8179 x 10 -15 9 . Let λ e be the Compton wavelength of the electron, 2.426 x 10 -12 m 10 . This topology is depicted in Fig 1A . It is the simplest knot, the unknot, twisted such that it forms two loops that are separated according to 2r e * sin θ. This general topology has been previously used to describe the electron as point-particles moving through curved space [START_REF] Hu | The nature of the electron[END_REF][START_REF] Williamson | Is the electron a photon with toroidal topology[END_REF] .

To this wire add charge equal to -2e, twice the expected amount. This charge will move through the wire at the speed of light, c, creating a current equal to,

I = qv l = - 2ec λ e (2)
In an external magnetic field this solenoid will experience torque proportional to the current in the wire [START_REF] Griffiths | Introduction to Electrodynamics[END_REF] . The magnetic dipole moment of a macroscopic current loop is given by,

µ = nIA = -2 * 2ec λ * πr 2 (3)
As r = λ 4π for this wire this formula simplifies to:

µ = -ecr = - ecλ e 4π (4)
This is exactly equivalent in value to the Bohr magneton, µ B = -9.274 x 10 -24 J/T 1 . Notably, this derivation of the Bohr magneton does not require a consideration of angular momentum, as the solenoid is assumed to be at rest.

Opposing forces

The magnetic field at the end of the loops can be similarly calculated as the field at the end of a solenoid, B = nµ 0 I 2d , where d is the distance between the loops 3 . To maintain a constant magnetic field regardless of position around the solenoid we will assume that the charge distribution varies by sin θ,

B = nµ 0 I * sin θ 2r e * sin θ = - µ 0 ec λ e r e (5)
This field is 1.1093 x 10 11 T, well above the Schwinger limit for a magnetic field (4.41 x 10 9 T). [START_REF] Sauter | Über das Verhalten eines Elektrons im homogenen elektrischen Feld nach der relativistischen Theorie Diracs[END_REF][START_REF] Schwinger | On Gauge Invariance and Vacuum Polarization[END_REF] While the current of a superconducting wire cannot vary by position while remaining at a constant charge density, a spinning charge distribution or oscillating standing wave of charge can [START_REF] Goedecke | Classically Radiationless Motions and Possible Implications for Quantum Theory[END_REF][START_REF] Long | Fundamentals of Acoustics[END_REF] . This suggests that while this superconducting wire may serve as a useful model, it is at best an incomplete description of reality.

Regardless, as the current moves around the ring it will experience centripetal force, F = mc 2 r . Counterbalance this force with the Lorentz force from a charge moving through a magnetic field of equivalent strength to that created at the ends of the solenoid, F = qvB 3 :

F = mc 2 r a = qvB = µ 0 e 2 c 2 λ a r e (6)
Rearranging for m gives,

m = µ 0 e 2 4πr e (7)
This is the rest mass of the electron.

This model also provides a geometric derivation of the fine structure constant [START_REF] Sommerfeld | Zur Quantentheorie der Spektrallinien[END_REF] . It is the ratio between the solenoid's diameter and half its height [START_REF] Hu | The nature of the electron[END_REF] ,

α = r e 2r = 2πr e λ e (8)
By inserting ( 8) into ( 7) and generalizing, we arrive at the previously established formula for the relationship between the rest mass of any particle and its Compton wavelength [START_REF] Arthur | A Quantum Theory of the Scattering of X-rays by Light Elements[END_REF] :

mλ = µ 0 e 2 2α = h c (9)

The anomalous magnetic moment

To account for the anomalous magnetic moment of the electron consider electrostatic repulsion equal to,

F e = e 2 16π 2 ε 0 r 2 (10)
Incorporating this into (7) gives a modified radius, r a :

m e c 2 r a = µ 0 e 2 c 2 λ a r e + e 2 16π 2 ε 0 r 2 a (11)
As r e = αλ 2π and λ a = 4πr a ,

m e c 2 r a = µ 0 e 2 c 2 8παr 2 a + e 2 16π 2 ε 0 r 2 a (12)
By rearranging for r a and incorporating the relationship c 2 = 1 µ 0 ε 0 this becomes,

r a = µ 0 e 2 8πm e * ( 1 α + 1 2π ) (13)
Incorporating this new radius into (4),

µ e = -ecr a = -ec * λ e + r e 4π (14)
This gives a µ e = -9.2847810029 x 10 -24 . While this is close to the experimental value of -9.2847647043 x 10 -24 J/T, [START_REF]CODATA Value: Electron magnetic moment[END_REF] the discrepancy between these values will require further study.

3 The proton and neutron

A superconducting trefoil knot

To determine the magnetic moment of the proton and neutron, we begin with the simplest prime knot after the unloop, the trefoil knot. This can be mathematically described by,

x = sin θ + 2 * sin 2θ y = cos θ -2 * cos 2θ z = -3 * sin 3θ (15)
Create an ethereal, infinitely thin wire with no electrical resistance along this path. Let the length of this wire equal the Compton wavelength of the proton, λ p = 1.3214 x 10 -15 m, and the charge of the wire equal to -2e [START_REF]CODATA Value: Proton magnetic moment[END_REF] . Contort this knot to match that of the electron, ie. such that it consists of two loops that are at most 2 * r p apart, where r p = αλ p 2π . Because the wire is ethereal it may freely superimpose with itself, creating cross points with zero displacement. This wire is depicted in Fig. 1B, and may be described by,

x = r * sin 2θ y = r * cos 2θ z = r p * sin 3θ (16)
As this trefoil consists of two circular loops, the magnetic moment is µ = nIA = -ecr. This is exactly equal in magnitude to the nuclear magneton [START_REF]CODATA Value: Nuclear magneton[END_REF] .

Assume that as with the electron the charge distribution varies according to sin θ, and that the magnetic field is reduced when ∆z r p * sin θ. Because z = r p * sin 3θ the average magnetic field along the z-axis is then,

B net = B max * 1 - π 2 0 ||sin (θ + u)| -|sin (3θ)|| dθ ( 17
)
Where u determines the phase of the charge distribution and B max is the magnetic field of an unloop with equivalent wavelength. This yields two maxima: one at u = 0,

B net = 1 -4 3 √ 2 -1 ≈ 0.447715B max and another at u = π 2 , B net = 3 + 2 √ 2 /3 - 4 3 4 + 2 √ 2 ≈ 0.458641B max .
As the magnetic field weakens the centripetal force will exceed the Lorentz force, elongating the path of the wire at each cross point. This will in turn reduce the curvature between cross points, weakening the centripetal force in these regions and giving the overall structure three-fold symmetry. The path of the wire is therefore, and a = B max B netb. This topology is depicted in Fig. 1C. The cross-section in the xy plane has been previously inferred from deep inelastic scattering between electrons and nucleons [START_REF] Breidenbach | Observed Behavior of Highly Inelastic Electron-Proton Scattering[END_REF][START_REF] Bloom | High-Energy Inelastic e -p Scattering at 6°and 10°[END_REF] , and the arms are commonly classified as up and down quarks in current representations of the proton and neutron 2,22-24 .

x =

Magnetic moments of the proton and neutron

To calculate the magnetic moments of the structures described by u = 0 and u = π 2 we begin with the formula for the magnetic moment of a solenoid, µ = nIA = -ecr.

To derive the magnetic moment for u = 0 bisect the structure. Place an axis of rotation along the x-intercept and integrate along the y-axis, multiplying the y-coordinate by cos θ to take into account the variation in charge density as a function of position (Fig. 2A). When r = λ n , the Compton wavelength for the neutron 1 , this value is -9.65833 x 10 -27 J/T, which compares favourably to the experimentally measured magnetic moment for the neutron, µ n = -9.66236 x 10 -27 J/T. [START_REF]CODATA Value: Neutron magnetic moment[END_REF] . Unlike this superconducting wire the neutron has no net charge, again indicating that this model, while useful, is fundamentally aphysical. To determine the magnetic moment of the structure described by u = π 2 we break the structure into two subcomponents. For the first, the axis of rotation is as far as possible from the middle cross point, such that it intersects the two side cross points (Fig. 2B). The displacement needed for this may be calculated by,

d = sin( π 6 ) * a * sin( π 2 ) + b = 1.09001963155 (20)
For the second component we place the central anti-node on the axis of rotation. This occurs at d = 0.

We then integrate both components along the x-axis,

µ = -ecr * π π 3 sin(θ - π 6 ) * sin(2θ + π 6 ) * (a * sin(6θ + π 2 + b) -1.0900176963155 dθ = 3.00938374141 * ecr (21)
For the first component, and,

µ = -ecr * 2 π 6 0 sin(θ - π 6 ) * sin(2θ + π 6 ) * (a * sin(6θ + π 2 + b) dθ = 0.312807503321 * ecr (22)
For the second component.

The net axis rotation is the weighted average of these two contributions,

d
When r is the Compton wavelength for the proton, the magnetic moment is 1.405825060 x 10 -26 J/T, approximately 99.66% that of the recorded magnetic moment for the proton, 1.410606797 x 10 -26 J/T. [START_REF]CODATA Value: Proton Compton wavelength[END_REF] .

Conclusion

In this work I have shown that it is possible to derive the magnetic moment of electron and the fine structure constant from a simple geometric model of a closed two-loop solenoid, and the magnetic moments of the proton and neutron from a similar wire folded into a trefoil knot. Inherent in these derivations is a charge distribution that varies based on position within wire, implying that these magnetic moments may be better represented as the result of oscillating standing waves or rotating charges. Work in this vein will be reported in due course. Bauer, J J Beatty, V I Belousov, J

Figure 1 :

 1 Figure 1: The paths described by equations A) (1), B) (16) and C) (18). Images are shown to scale, with the Z axis elongated for clarity. The magnetic moments of A and C correspond to the electron and proton/neutron, respectively.

  r * sin 2θ * (a * sin(6θ + b) y = r * cos 2θ * (a * sin(6θ + b) z = r p * sin 3θ

Figure 2 :

 2 Figure 2: Axes of rotation for the A) neutron and B) proton when the particles are considered as superconducting wires. The wire loops are shown as cross-sections in the xy plane. The charge density is approximated by thick lines, while the cross points are shown with A) orange or B) blue circles.

  net = d minor * µ minor * ecr µ minor + µ ma jor + d ma jor * µ ma jor * ecr µ minor + µ ma jor = 0.987384910135 (23) Incorporate d net into the combined equation, offset by π 6 to maintain a consistent phase over the entire integral: µ = -ecr * sin(6θ + π 2 + b) -0.987384910135 dθ = 2.783380027 * ecr

  Beringer, A Bettini, O Biebel, K M Black, E Blucher, R Bonventre, V V Bryzgalov, O Buchmuller, M A Bychkov, R N Cahn, M Carena, A Ceccucci, A Cerri, R Sekhar Chivukula, G Cowan, K Cranmer, O Cremonesi, G D'Ambrosio, T Damour, D de Florian, A de Gouvêa, T DeGrand, P de Jong, S Demers, B A Dobrescu, M D'Onofrio, M Doser, H K Dreiner, P Eerola, U Egede, S Eidelman, A X El-Khadra, J Ellis, S C Eno, J Erler, V V Ezhela, W Fetscher, B D Fields, A Freitas, H Gallagher, Y Gershtein, T Gherghetta, M C Gonzalez-Garcia, M Goodman, C Grab, A V Gritsan, C Grojean, D E Groom, M Grünewald, A Gurtu, T Gutsche, H E Haber, Matthieu Hamel, C Hanhart, S Hashimoto, Y Hayato, A Hebecker, S Heinemeyer, J J Hernández-Rey, K Hikasa, J Hisano, A Höcker, J Holder, L Hsu, J Huston, T Hyodo, Al Ianni, M Kado, M Karliner, U F Katz, M Kenzie, V A Khoze, S R Klein, F Krauss, M Kreps, P Križan, B Krusche, Y Kwon, O Lahav, J Laiho, L P Lellouch, J Lesgourgues, A R Liddle, Z Ligeti, C-J Lin, C Lippmann, T M Liss, L Littenberg, C Lourenço, K S Lugovsky, S B Lugovsky, A Lusiani, Y Makida, F Maltoni, T Mannel, A V Manohar, W J Marciano, A Masoni, J Matthews, U-G Meißner, I-A Melzer-Pellmann, M Mikhasenko, D J Miller, D Milstead, R E Mitchell, K Mönig, P Molaro, F Moortgat, M Moskovic, K Nakamura, M Narain, P Nason, S Navas, A Nelles, M Neubert, P Nevski, Y Nir, K A Olive, C Patrignani, J A Peacock, V A Petrov, E Pianori, A Pich, A Piepke, F Pietropaolo, A Pomarol, S Pordes, S Profumo, A Quadt, K Rabbertz, J Rademacker, G Raffelt, M Ramsey-Musolf, B N Ratcliff, P Richardson, A Ringwald, D J Robinson, S Roesler, S Rolli, A Romaniouk, L J Rosenberg, J L Rosner, G Rybka, M G Ryskin, R A Ryutin, Y Sakai, S Sarkar, F Sauli, O Schneider, S Schönert, K Scholberg, A J Schwartz, J Schwiening, D Scott, F Sefkow, U Seljak, V Sharma, S R Sharpe, V Shiltsev, G Signorelli, M Silari, F Simon, T Sjöstrand, P Skands, T Skwarnicki, G F Smoot, A Soffer, M S Sozzi, S Spanier, C Spiering, A Stahl, S L Stone, Y Sumino, M J Syphers, F Takahashi, M Tanabashi, J Tanaka, M Taševský, K Terao, K Terashi, J Terning, R S Thorne, M Titov, N P Tkachenko, D R Tovey, K Trabelsi, P Urquijo, G Valencia, R Van de Water, N Varelas, G Venanzoni, L Verde, I Vivarelli, P Vogel, W Vogelsang, V Vorobyev, S P Wakely, W Walkowiak, C W Walter, D Wands, D H Weinberg, E J Weinberg, N Wermes, M White, L R Wiencke, S Willocq, C G Wohl, C L Woody, W-M Yao, M Yokoyama, R Yoshida, G Zanderighi, G P Zeller, O V Zenin, R-Y Zhu, Shi-Lin Zhu, F Zimmermann,

Acknowledgements

I wish to thank Drs. Ingo Salzmann, Serhat Alagöz, Cameron Skinner, and Saurabh Maiti for helpful discussions, John Duffield, whose blog helped formalize some of my ideas, and Dr. Lisa Freeman, for always being willing to listen.


Alin Watanabe

Bostan

Watanabe Ren- Diconti

Krattenthaler Conca

Watanabe Alin

A short proof of a non-vanishing result by Conca, Krattenthaler and

Keywords: Recursive sequences, binomial coefficients, generating functions

.

In their paper Regular sequences of symmetric polynomials [START_REF] Conca | Regular sequences of symmetric polynomials[END_REF], Aldo Conca, Christian Krattenthaler and Junzo Watanabe needed to prove, as an intermediate result, the fact that for any h ≥ 1, the rational number

⌊h/3⌋ b=0 (-1) h-b h -b h -b 2 b 2 3
b is non-zero, except for h = 3. The proof in [START_REF] Conca | Regular sequences of symmetric polynomials[END_REF]Appendix, performs a (quite intricate) 3-adic analysis. In this note, we propose a shorter and elementary proof, based on the following observation.

Theorem 1. For any h ≥ 1, consider the polynomials

a h := ⌊h/3⌋ b=0 (-1) h-b h -b h -b 2b U b ∈ Q[U ]
and s h := h • a h . Then, the sequence (s h ) h≥1 satisfies the linear recurrence

(1) s h+3 + 2 s h+2 + s h+1 = U • s h , for all h ≥ 1. Proof. Using h/(h -b) = 1 + b/(h -b) and 2b • h-b 2b = (h -b) • h-b-1 2b-1
yields the additive decomposition s h = p h + q h , where

p h := ⌊h/3⌋ b=0 (-1) h-b h -b 2b U b and q h := 1 2 • ⌊h/3⌋ b=0 (-1) h-b h -b -1 2b -1 U b .
( * ) Indirizzo dell'A.: Inria, Université Paris-Saclay, France. E-mail: alin.bostan@inria.fr

It is thus enough to prove that both (p h ) h≥1 and (q h ) h≥1 satisfy recurrence (1). We prove this for (p h ) h≥1 , the proof for (q h ) h≥1 being similar. Extracting the coefficient of U n on both sides of (1) with (s h ) replaced by (p h ) is equivalent to

h + 3 -n 2 n -2 h + 2 -n 2 n + h + 1 -n 2 n = h + 1 -n 2 n -2
and this identity is an immediate consequence of the Pascal triangle rule. □ Corollary 2. For any h ≥ 1, the rational number

⌊h/3⌋ b=0 (-1) h-b h -b h -b 2 b 2 3 b is non-zero, except for h = 3.
Proof. With previous notation, we need to prove that a h (2/3) = 0 if and only if h = 3. By Theorem 1, the sequence

(u h ) h≥1 := 3 h-1 • h • a h (2/3) h≥1 = (-1, 3, 0, -45, 324, . . .)
satisfies the linear recurrence relation (2) u h+3 + 6 u h+2 + 9 u h+1 = 18 u h , for all h ≥ 1.

It is clearly enough to prove that u h = 0 if and only if h = 3. First, the terms u h are all integers, by induction. Recurrence (2) shows that u h+3 and u h+1 have the same parity for all h ≥ 1; since u 2 = 3, this implies that u 2h is an odd integer, and in particular it is nonzero, for all h ≥ 1. It remains to consider the odd subsequence (v h ) h≥1 := (u 2h-1 ) h≥1 = (-1, 0, 324, 5508, 2916, . . .).

From (2) it follows that the sequence (v h ) h≥1 satisfies the recurrence relation

(3) v h+3 -18 v h+2 + 297 v h+1 = 324 v h for all h ≥ 1.
The same recurrence is also satisfied with (v h ) h≥1 replaced by the sequence (w h ) h≥1 := (v h /4) h≥3 = (81, 1377, 729, -369603, . . .). In particular, w h+3 and w h+1 have the same parity for all h ≥ 1, hence w h is odd for any h ≥ 1. It follows that v h is nonzero for all h ≥ 3, which concludes the proof. □ Remark 3. An equivalent, equally simple, but slightly more "conceptual" proof of Theorem 1 is expressed in terms of generating functions.

a 2b U b z a = 1 -z (1 -z) 2 -U z 2 , a,b a -1 2b -1 U b z a = U z 2 (1 -z) 2 -U z 2 ,
and finally substitutes successively a ← h -b, z ← -z, U ← U z; this yields

h≥1 s h z h = z + 1 (1 + z) 2 -U z 3 -1 + 1 2 • U z 3 (1 + z) 2 -U z 3 = z + 1 + U z 3 /2 (1 + z) 2 -U z 3 -1.
Recurrence (1) is now read off the denominator of the last rational function.

Remark 4. We leave it as an open problem to prove that the polynomials a h (U ) and s h (U ) are irreducible in Q[U ] for all h ≥ 3. (Computer calculations show that this holds for 3 ≤ h ≤ 10000.) If true, this would imply a generalization of Corollary 2.

  One starts with the Pascal triangle rule in its "generating function" form a,b a b U b z a = 1/(1 -(1 + U )z), then extracts odd and even parts (with respect to U ) from it, a,b


Nathalie Cotte

Anette Helle

Pedersen

H A Pedersen
email: helle.pedersen@obs.ujf-grenoble.fr

TOR Working Group

S Gregersen

A Berthelsen

H Thybo

K Mosegaard

T Pedersen

P Voss

( Denmark

L B Pedersen

R G Roberts

H Shomali

R Kind

G Bock

J Gossler

K Wylegalla

W Rabbel

I Woelbern

M Budweg

H Busche

M Korn

S Hock

A Guterch

M Grad

M Wilde-Pio

M Zuchniak

( Poland

J Plomerova

Czech Rep

J Ansorge

E Kissling

R Arlitt

F Waldhauser

P Ziegler

Sharp contrast in lithospheric structure across the Sorgenfrei Tornquist Zone as inferred by Rayleigh wave analysis of TOR1 project data

Keywords: Lithospheric structure, Sorgenfrei -Tornquist Zone, Rayleigh wave analysis, TOR1 project data

1 1. Introduction
All previous studies (see below) agree in proposing significant lateral variations in the structure of the lithosphere across the Tornquist Zone. The Sorgenfrei -Tornquist Zone (STZ) divides Central Europe (Phanerozoic) from Fennoscandia (Proterozoic), whereas Central Europe and Eastern Europe are separated by the Teysseire -Tornquist Zone (TTZ). These two branches of the Tornquist Zone join between Germany and Sweden (e.g., [START_REF] Berthelsen | Mobile Europe[END_REF][START_REF] Guterch | Deep structure of the earth's crust in the contact zone of the Paleozoic and Precambrian platforms in Poland (Tornquist -Teisseyre zone)[END_REF][START_REF] Guterch | Crustal structure of the transition zone between Precambrian and Variscan Europe from new seismic data along LT-7 profile (NW Poland and eastern Germany)[END_REF].

The STZ is the north-eastern border of the Tornquist Fan. This region is delimited to the south -west by the Trans-European Fault (TEF) (e.g., [START_REF] Berthelsen | Mobile Europe[END_REF] and includes Denmark. The Tornquist Fan has an ambiguous origin: on the one hand, the STZ marks a sharp transition in the tectonic properties of the crust from the Baltic Shield (35 -45 km thick) to the Tornquist Fan (f32 km) (e.g., [START_REF] Tryggvason | A two-dimensional seismic velocity model across the transition zone between the Baltic Shield and the North German Basin-the EUGENO-S profile 1 revisited[END_REF]; on the other hand, however, the basement of the Tornquist Fan is similar to that of the Baltic Shield (e.g., [START_REF] Berthelsen | Mobile Europe[END_REF]. Compounding the ambiguity is the fact that lateral variations in the structure of the lithosphere are not as well constrained as crustal ones, as no dense, passive seismic experiment have been carried out in the area. Crustal models have been determined by reflection and refraction seismic studies such as FENNO-LORA [START_REF] Guggisberg | A two-dimensional velocity model for the lithosphere beneath the Baltic Shield and its possible tectonic significance[END_REF], European GeoTraverse (1992), EUGENO-S Working [START_REF] Group | Crustale structure and tectonic evolution of the transition between the Baltic Shield and the North German Caledonides (the EUGENO-S Project)[END_REF] a n dB A B E L ,a n da r ew e l lr e p o r t e di nt h e literature [START_REF] Guggisberg | Crustal structure of the Fennoscandian Shield: a traveltime interpretation of the long-range FENNOLORA seismic refraction profile[END_REF][START_REF] Thybo | A seismic velocity model along the EGT profile from the North German Basin into the Baltic Shield[END_REF][START_REF] Berthelsen | Mobile Europe[END_REF]BABEL Working Group, 1993;[START_REF] Tryggvason | A two-dimensional seismic velocity model across the transition zone between the Baltic Shield and the North German Basin-the EUGENO-S profile 1 revisited[END_REF][START_REF] Thybo | Interpretation in statu nascendi of seismic wide-angle reflections based on EUGENO-S data[END_REF].

It is now well known that strong lateral variations exist in the lithosphere between the ''old'' Eastern Europe and the ''young'' Central Europe. Several surface wave studies have focused on this structure. [START_REF] Snieder | Large-scale waveform inversions of surface waves for lateral heterogeneity: 2. Application to surface waves in Europe and the Mediterranean[END_REF] determined a very sharp transition in lithospheric structure from one side of the TTZ to the other, between the depths of 100 and 200 km. [START_REF] Zielhuis | The deep seismic expression of an ancient plate boundary in Europe[END_REF] determined the seismic structure of the TTZ to a depth of at least 140 km, below which the contrast in velocity from a side to the other of the TTZ is smaller. Subsequently, [START_REF] Marquering | Shear-wave velocity structure beneath Europe, the northeastern Atlantic and western Asia from waveform inversion including surface-wave mode coupling[END_REF] showed the existence of large lateral variations in lithospheric structure in the vicinity of the TTZ between the depths of 80 and 140 km, representing a sharp transition. [START_REF] Alsina | Constraints on the velocity structure beneath the Tornquist -Teisseyre Zone from beam-forming analysis[END_REF] determined the direction of incident surface waves on the TTZ and showed that large lateral refractions occur on this zone. Using body waves, [START_REF] Schweitzer | Blockage of regional seismic waves by the Teisseyre -Tornquist zone[END_REF] showed how the TTZ presents a blockage for P n and P g rays perpendicular to it. His observation of shadow areas leads to the proposal of a LVZ extending to a depth of 200 km. There is therefore no doubt that the TTZ is a major discontinuity down to depths of at least 200 km.

North of the TTZ, several studies have provided information about the thickness of the Baltic Shield and showed how it varies from one side of the STZ to the other. Using P-wave residuals, [START_REF] Husebye | Seismic mapping of the Fennoscandian lithosphere and asthenosphere with special reference to the Oslo graben[END_REF] and [START_REF] Husebye | The upper mantle seismic heterogeneities beneath Fennoscandia[END_REF] demonstrated that the Baltic Shield was characterized by high velocities down to depths of 250-300 km. From P-and S-wave refraction, [START_REF] Sacks | Lithosphere thickness beneath the Baltic Shield[END_REF] determined a Baltic Shield thickness of 250 F 15 km at 2-3j from the NORSAR array. In 1990, Dost showed that the lithosphere -asthenosphere boundary does not appear beneath the Baltic Shield. Using the fundamental and first two higher modes of Rayleigh waves, [START_REF] Calcagnile | Deep structure of Fennoscandia from fundamental and higher mode dispersion of Rayleigh waves[END_REF] determined a LVZ extending between 140 and 350 km depth beneath Sweden, and between 220 and 290 km depth under the Baltic Sea. In 1994, Pedersen et al. employed an analysis of Rayleigh waves to determine that such a LVZ does not appear beneath Norway but exists beneath Denmark, between the depths of 120 and 220 km. Finally, by correlating lithospheric thicknesses with heat flux, [START_REF] Pollack | On the regional variation of heat flow, geotherms and lithospheric thickness[END_REF] argued for the existence of a 200-km thick shield.

All these studies, except the one of [START_REF] Calcagnile | Deep structure of Fennoscandia from fundamental and higher mode dispersion of Rayleigh waves[END_REF], agree in proposing a thick lithosphere for the Baltic Shield although disagreements remain with respect to the inferred value of the thickness and shear-wave velocities. In the south, the transition is expected to be sharp across the TTZ, while the sharpness is more disputed in the north across the STZ, especially because rocks of Proterozoic age have been found south -west of the STZ, and no seismic data on the area have had sufficient resolution to study the STZ on a lithospheric scale. This present study is thus devoted to the determination of the lithospheric structures across the STZ and the TEF.

The TOR1 seismic experiment and broad-band data

The TOR1 experiment (''Teleseismic TOmography Experiment across the TORnquist Zone'') took place in 1996 -1997 with the goal of inferring the structure of the lithosphere across the Tornquist Zone by using tomography, receiver-function, anisotropy and surface wave studies [START_REF] Gregersen | Important findings expected from Europe's largest seismic array[END_REF]. It represented a multinational effort (the participating countries and workers are listed at the beginning of the paper) and was part of EUROPROBE program.

One hundred twenty seismometers were installed from Go ¨ttingen (Germany) to Stockholm (Sweden) during the period covering September 1996 to May 1997. The location of the seismic stations is shown in Fig. 1 (modified after [START_REF] Gregersen | Important findings expected from Europe's largest seismic array[END_REF]. Among the seismic stations, 38 were broadband stations either from permanent networks or from the temporary experiment, which records are used for studying surface waves.

Six mini-arrays composed of three broadband stations are defined, some on each side of the STZ: two in Sweden (named S1 and S2), two in Denmark Fig. 1. Location of the short period stations (small grey points) and broadband stations (large black points) operating during the TOR1 experiment, from September 1996 to May 1997 (modified after [START_REF] Gregersen | Important findings expected from Europe's largest seismic array[END_REF]. Dashed circles represent the mini-arrays we used in this study.

(D1 and D2) and two in Germany (G1 and G2) (see Fig. 1). The stations are installed in triangles of 40 -50 km width. The positioning of the mini-arrays permits the study of variations of lithospheric structure on the small scale of a few dozens of kilometers. Note that D2 and D1 arrays have two stations in common, but do not overlap in space. Phase velocities measured within these two arrays therefore correspond to different structures. One more array was available in Denmark, but could not be used due to GPS timing errors.

We selected 99 events for this study, with a magnitude greater than 5.1. The epicentral distance ranges from 15j to 155j and the azimuth coverage is very good, as shown in Fig. 2, as there is only a small gap between 310j and 345j. We discarded events for which the epicentral distance was larger than 155j to avoid problems due to multipathing and focusing/defocusing effects which can occur when the source is close to the antipodes. Data were filtered between 5 and 200 s and corrected from the instrument response.

Fig. 3 shows 26 seismograms recorded on the vertical component by temporary and permanent broadband stations for a source located in the Atlantic ocean. The incident surface waves thus arrive from the south -west. The seismograms are all sorted by epicentral distance and the amplitude scale is the same for all of them. The first phase, recorded approximately at 1100 s by the nearest station to the source and at 1200 s by the further one, is the direct shear-wave. The main phase in amplitude is the Rayleigh wave, between 1700 and 2200 s. We note the good signal-to-noise ratio and the good coherence between records. Unfortunately, the signal-to-noise ratio of the horizontal components does not allow us to consider Love waves in the analysis. This is due to the thickness of the sediments as well as noise from the sea affecting most of the stations.

Phase velocity measurements for Rayleigh waves

This section is devoted to phase velocity measurements of Rayleigh waves, both within mini-arrays and along long profiles of more than hundred kilometers. Within the mini-arrays, we used the method described by [START_REF] Cotte | Off-great circle propagation of intermediate period surface waves as observed on a dense array in the French Alps[END_REF]. In this paper, we presented the way to measure the phase velocity within arrays whose size is small compared to the wavelength of the surface waves. This analysis is possible thanks to the high coherence between stations. Firstly, we measure the arrival direction of the incident wave. For each frequency, we measure the time delay for all the pairs of stations using the phase of the crossspectrum. Knowing the distances between stations, the time delays are inverted to obtain the slowness vector. We directly get the arrival direction of the incident wave. Secondly, we report on a diagram the time delays as a function of the interstation distance corrected from the off-great circle deviation as determined by the slowness vector. Then, a linear regression through the points, and passing by the origin gives the phase velocity as the inverse of the slope of the line.

Large distances between stations are not necessary as long as the coherency between records is very high, so the phase velocity measurement within a small array is better adapted for studying laterally heterogeneous media than a measurement along a long profile, more than 100 km long, which neglects the Fig. 3. Example of seismograms recorded on the vertical component by temporary and permanent broadband stations for the TOR1 experiment for an event located in the mid-Atlantic ridge (December 12, 1996). The seismograms are filtered between 5 and 200 s and corrected for instrument response. They are all sorted by epicentral distance and we use the same amplitude scale in all cases. The coherency between signals is high, so phase velocity measurements based on coherency can be performed. lateral heterogeneities. Measuring the phase velocity within a mini-array presents the advantage to take the observed off-great circle deviations into account. [START_REF] Cotte | Off-great circle propagation of intermediate period surface waves as observed on a dense array in the French Alps[END_REF] showed that these deviations can reach up to 30j at 30 s period in the French Alps. It is however necessary to apply severe selection criteria to the data, and eliminate all records with an unstable phase. Nonetheless, for covering a large area of study we also measure the phase velocity along long profiles (several hundreds of kilometers). For this regional approach, we present phase velocities obtained along profiles averaged within each major tectonic unit.

Fig. 4a andb shows the dispersion curves measured for the Rayleigh wave in Denmark (D1 and D2 arrays) and Germany (G1 and G2 arrays), with respectively three, eight, four and five events. We did not succeed in measuring the phase velocities using the arrays located in Sweden because large amounts of data were missing, the data were very noisy, and the frequency band was too narrow for studying the lithospheric structure. The grey area is the confidence interval that we used in the inversion, determined by all our measurements. The black dots show the average value of the measured phase velocity.

In Fig. 4a, we can see that the phase velocity increases rapidly when the period varies from 20 to 35 s, and more slowly when the period varies from 35 to 100 s for the D1 array measurements. For the D2 array, a large scatter is observed in the measurements for periods shorter than 50 s and longer than 90 s. We thus decided to discard measurements lying outside 50-90 s. Within this frequency range, the phase velocity increases slightly with period and the confidence interval in velocity is small. For the G1 array (Fig. 4c), the phase velocity steadily increases when the period varies from 30 to 100 s. At 80-85 s period, the confidence interval is large, up to F 1.5 km/s. Finally, the G2 array presents an important increase of the velocity when the period varies from 25 to 30 s, a constant velocity when the period varies from 30 to 70 s, and an increasing velocity for greater periods, reaching 4.4 km/s at 100 s period.

We also measured the phase velocity along profiles for which the interstation distance was greater than the wavelength at 30 s period, i.e. 100-120 km, using the Wiener filtering method [START_REF] Wiener | Time Series[END_REF]; see also [START_REF] Nakanishi | Phase velocity and Q of mantle Rayleigh waves[END_REF][START_REF] Taylor | Measurement of interstation phase and group velocities and Q using Wiener filtering[END_REF]Hwang and Mitchell, 1986). Fig. 5 shows all the profiles we used. We divided the area into three different zones which correspond to three tectonic units: Sweden (Baltic Shield, dark grey profiles), Denmark (between STZ and TEF, black profiles), and Germany (Phanerozoic Europe, light grey profiles). We respectively Fig. 5. Profiles of broadband stations used for measuring the phase velocity in Sweden (dark grey profiles), Denmark (black profiles), and Germany (light grey profiles) which correspond to three different tectonic areas. made 9, 11 and 47 measurements along profiles, using 7, 6 and 28 different events, some of the measurements corresponding to a same event, but events are always selected so that the theoretical path and the station pairs present a misalignment lower than 5j. This precaution is for minimizing the effect due to offgreat circle propagation as we can not correct the phase velocity of the great-circle deviation. We also verified that events from different directions yield approximately the same dispersion curves.

Results of phase velocity measurement along profiles are shown in Fig. 6. We can see in the shape of the dispersion curves that lithospheric models for the three areas must be different. For short periods, the phase velocity is low in Germany (3.45 km/s at 20 s) as compared to Denmark and Sweden (3.65 and 3.60 km/s at 20 s). For intermediate periods (30 -50 s), Denmark and Germany have same phase velocities, while they are up to 0.2 km/s larger for Sweden. Finally, for periods greater than 70 s, Sweden is characterized by larger phase velocity values (up to 4.5 km/s at 90 s) than Germany or Denmark (4.15 and 4.05 km/s). The dispersion curves within the arrays are different to the ones along the long profiles. The reason is that long profiles average the structure between the stations, while within a small array it is more obvious to work on a 1D model, assuming that the lateral heterogeneities of wavelength of 50-300 km are large compared to the array aperture (40 -50 km). Therefore, measurements along long profiles and within arrays must be inverted separately and will give complementary information on the lithospheric structures.

Inversion of dispersion curves: construction of lithospheric models

Using the seven dispersion curves of Figs. 4 and6, we performed inversions to determine the corresponding lithospheric models in shear-wave velocity. We used the two-step method suggested by [START_REF] Shapiro | Surface-wave propagation across the Mexican Volcanic Belt and the origin of the long-period seismic-wave amplification in the valley of Mexico[END_REF]. The first step consists in using a linearized inversion scheme as the one of [START_REF] Herrmann | Computer programs in seismology[END_REF]. We get a simple lithospheric model, which is obtained by a root mean square fitting of the phase velocity dispersion curve. From this model, the second step consists in applying a random change either in velocity or in thickness for all the different layers. If the new model has a theoretical dispersion curve, which falls within our confidence interval, then the model is kept as a new solution and a new change is applied to it. Otherwise, we reconsider the last solution and apply a new change. This Monte-Carlo method allows us to explore the solutions for describing the set of possible models for which the dispersion curve fits within our confidence interval. By testing thousands of different models, we finally get a large set of solutions. The advantage is that we can explore all possible solutions and therefore avoid selecting a model corresponding to a secondary minimum in the inversion. We allow changes in the model to be up to 0.2 km/s in velocity for each layer, and of a few kilometers for the interface depth (the deeper is the interface, the bigger can be the change). We only accepted models for which crustal thicknesses and uppermost mantle velocities were coherent with results from refraction and/or reflection seismic profiling (e.g. [START_REF] Gregersen | Interpretation from explosion seismograms of crustal inhomogeneities in Statu Nascendi[END_REF][START_REF] Tryggvason | A two-dimensional seismic velocity model across the transition zone between the Baltic Shield and the North German Basin-the EUGENO-S profile 1 revisited[END_REF][START_REF] Thybo | Interpretation in statu nascendi of seismic wide-angle reflections based on EUGENO-S data[END_REF][START_REF] Pedersen | Project TOR: deep lithospheric variation across the Sorgenfrei -Tornquist Zone, Southern Scandinavia[END_REF]. The limits imposed were for crustal thicknesses 38 F 2 km (Sweden), 32 F 2 km (Denmark) and 30 F 2 km (Germany). The imposed upper mantle velocities were 4.50 -4.80 km/s (Sweden), 4.50 -4.70 km/s (Denmark) and 4.40-4.70 km/s (Sweden). Below this layer, no conditions are imposed on the S-wave velocities. Fig. 7 shows in grey lines all the set of models found by inversion for the arrays. They are shearwave velocity models expressed as a function of the depth and are reliable to depths of 200 km. For greater depths, we do not have the necessary resolution that would allow us to constrain the models. For all depths, we determine both the mean velocity as the average for all the possible models (solid line) and the mean using the minimal and maximal velocities (dashed line) to show the values given by 'extreme' models. The mean model is not one particular solution of the inversion. We determine the velocity in the uppermost mantle as the mean value found within the first kilometers beneath the Moho. The velocity at the minimum in the asthenosphere is the smallest value found by averaging all the models. Determining the lithospheric thickness is less obvious as there is a great variability in the models. We defined the minimum lithospheric thickness as the depth below which two-thirds of the models have an inverse gradient and the maximum lithospheric thickness as the depth where the velocity equals the minimum velocity of the asthenosphere + 10% of its standard deviation. The average lithospheric thickness is determined as the average of these two values. We emphasize that having a rather smooth mean model makes the determination of the lithospheric thickness somewhat difficult, but this is a problem of real earth structure and not of the inversion procedure.

In Fig. 7, we can see that beneath the G1 and G2 arrays, the lithosphere-asthenosphere boundary is inferred to be at depths of 50 F 10 and 75 F 25 km, respectively. Shear-wave velocities in the upper mantle are respectively 4.43 F 0.06 and 4.57 F 0.08 km/s, and are 4.39 F 0.05 and 4.36 F 0.09 km/s at the minimum in the asthenosphere. For the G1 array, the large difference between the mean velocity (4.43 F 0.06 km/s) in the uppermost mantle and the value given by the two extreme models (4.54 F 0.16 km/s) shows that this velocity is not well determined. Beneath the D1 array, no lithosphere -asthenosphere boundary appears as the shear-wave velocity steadily increases with depth, to depths of at least 200 km. As the incident Rayleigh waves all arrive from northeast, we cannot exclude the possibility of a signature of the propagation in Sweden strongly affecting what we determine within the D1 array. Fifty kilometers to the south -west of this array, beneath the D2 array, we determine a lithosphereasthe-asthenosphere boundary located at the depth of 120 F 20 km, with shear-wave velocity of 4.60 F 0.04 km/s in the upper mantle and 4.45 F 0.09 km/s at the minimum in the asthenosphere.

The same inversion procedure was used for the dispersion curves measured along profiles in Germany, Denmark, and Sweden (see Fig. 6). Results for these three areas are presented in Fig. 8. Beneath Germany (Fig. 8a), the lithosphere is rather thin with an average thickness of 100 F 20 km, and shear-wave velocities of 4.54 F 0.03 and 4.46 F 0.08 km/s in the upper mantle and at the minimum in the asthenosphere, respectively. The lithospheric structure determined along long profiles is thicker than the one determined beneath the G1 and G2 arrays, but profiles cover a much greater area than the two arrays. Surprisingly, the velocity at the minimum in the asthenosphere is larger using all profiles than the ones found beneath the two arrays. We speculate that this is due to the fact that the velocity is determined as a greater depth as compared to the arrays, which might be correlated to the thicker lithosphere. Beneath Denmark (Fig. 8b), the lithosphere is 120 F 20 km thick Fig. 7. Lithospheric models determined by inversion of the Rayleigh wave dispersion curves for the D1 and D2 arrays in Denmark (a and b) and the G1 and G2 arrays in Germany (c and d). Grey lines show all the models solution of the inversion, while solid black line gives the mean velocity over all models and the dashed black line is the average between the minimal and maximal velocities. and the shear-wave velocities are almost the same as in Germany: 4.57 F 0.06 km/s in the upper mantle and 4.43 F 0.08 km/s at the minimum in the asthenosphere. In that case, the lithosphere thickness and velocities are the same as the ones determined beneath the D2 array. This may indicate that no large lateral heterogeneities are likely to exist beneath Denmark, apart from the variation between D1 and D2. Finally, we show the inversion for measurements along profiles in Sweden (Fig. 8c). This model does not present any lithosphere -asthenosphere boundary to depth of at least 200 km. No array analysis was available for this area. The shear-wave velocity determined in the upper mantle, of 4.66 F 0.03 km/s, is higher than the ones determined for Denmark or Germany.

Discussion and conclusion

In Table 1, we have summarized our results for the thicknesses of the lithosphere and the shear-wave velocities in the upper mantle that we determined under Sweden, Denmark and Germany by the inversion of Rayleigh wave dispersion curves. We propose in Fig. 9 a lithospheric model along the TOR1 profile. This model is constructed using the results from the four arrays, and outside them using the available information from the three average models from Sweden, Denmark and Germany.

North of the Sorgenfrei -Tornquist Zone, beneath the Baltic Shield in Sweden, no lithosphere -asthenosphere boundary is inferred to depths of at least 200 km. Our results are consistent with the ones of [START_REF] Husebye | Seismic mapping of the Fennoscandian lithosphere and asthenosphere with special reference to the Oslo graben[END_REF], [START_REF] Stuart | The upper mantle structure of the North Sea region from Rayleigh wave dispersion[END_REF], [START_REF] Husebye | The upper mantle seismic heterogeneities beneath Fennoscandia[END_REF], [START_REF] Sacks | Lithosphere thickness beneath the Baltic Shield[END_REF], [START_REF] Dost | Upper mantle structure under western Europe from fundamental and higher mode surface waves using the NARS array[END_REF], [START_REF] Pedersen | Changes in the lithospheric structure across the Sorgenfrei -Tornquist Zone inferred from dispersion of Rayleigh waves[END_REF] and [START_REF] Pollack | On the regional variation of heat flow, geotherms and lithospheric thickness[END_REF]. They are consistent with the result of [START_REF] Calcagnile | The lithosphere -asthenosphere system in Fennoscandia[END_REF][START_REF] Calcagnile | Deep structure of Fennoscandia from fundamental and higher mode dispersion of Rayleigh waves[END_REF] for the Danish area, but in disagreement on the structure beneath southern Sweden. However, due too a poor station coverage of the area prior the TOR experiment, direct comparison between their studies and ours is somewhat difficult. We also have no lithosphere -asthenosphere boundary appearing under the D1 array located just south -west of the Sorgenfrei -Tornquist Zone. Beneath Denmark, with the exception of the structure determined under the D1 array, the lithospheric thickness is 120 F 20 km. We observed no lateral variations in the depth of the lithosphere -asthenosphere boundary and only small variations for the shear -wave velocities in the upper mantle and at the minimum in the asthenosphere. Finally, beneath Germany we determined a very thin lithosphere of thickness smaller or equal to 100 F 20 km, in particular close to Denmark where the G1 and G2 arrays are located.

We thus conclude that the major discontinuity in the lithospheric structure across the Tornquist Fan is located between Denmark and Sweden, where the Sorgenfrei -Tornquist Zone lies at the surface. Indeed, within a few dozens of kilometers the lithospheric structure changes laterally very rapidly and the Sorgenfrei-Tornquist Zone is confirmed to be a major tectonic feature to depths of at least 200 km. As we have the same lithospheric structure beneath Sweden and beneath the D1 array located in Denmark, just south -west to the Sorgenfrei -Tornquist Zone, we conclude that the tectonic transition is sharp, dipping steeply to the south -west. The Baltic Shield extends under the D1 array, but does not appear under the D2 array, which is 50 km further from the STZ.

A second and smaller discontinuity is also determined across the Trans-European Fault located between Denmark and Germany, south of the Ringkbing Fyn High (Fig. 1). North of it, in Denmark, the lithosphere is 120 F 20 km thick and the shear-wave velocity at the minimum in the asthenosphere is approximately 4.43-4.45 km/s. South of it, in Germany, the thickness of the lithosphere is smaller, of 50 F 10 km under the G1 array, and the shear-wave velocity inferred to be 4.39 km/s at the minimum in the asthenosphere beneath the G1 array is rather low. Even if the Trans-European fault is a postulated one [START_REF] Berthelsen | The early (800 -300 Ma) crustal evolution of the off-shield region in Europe[END_REF], our results show significant contrasts in the lithospheric structure across its assumed location.

The only other previous surface wave study that specifically adresses the STZ [START_REF] Pedersen | Changes in the lithospheric structure across the Sorgenfrei -Tornquist Zone inferred from dispersion of Rayleigh waves[END_REF] showed a significant difference between Western Denmark and Eastern Norway. They determined a 120-km thick lithosphere beneath the former, and no lithosphere-asthenosphere boundary beneath the latter, which is consistent with our results. We improved the constraint of lateral variation by having much more stations and by performing array analysis on a small scale as compared to the wavelength of the studied Rayleigh waves, thereby limiting the contrast in lithospheric structure to the STZ. More recently, by performing P-wave tomography within the context of the TOR1 experiment, [START_REF] Arlitt | P-wave velocity structure of the lithosphere -asthenosphere system across the TESZ in Denmark[END_REF] showed that no lithosphere -asthenosphere boundary exists beneath the Baltic Shield to a depth of 200 km and that the lithosphere is 120 km beneath Denmark but otherwise relatively thin (50 km) beneath Germany. Note that they did not have a good resolution at this depth due to the limit of the ''crustal'' model (50 km thick) used for the travel time inversion [START_REF] Arlitt | 3-D crustal structure beneath the TOR array and effects on teleseismic wavefronts[END_REF]. Their results for the thickness of the lithosphere and the geometry of the tectonic feature beneath the Sorgenfrei-Tornquist Zone, dipping steeply to the south -west, are in good agreement with ours. On the other hand, [START_REF] Pedersen | Project TOR: deep lithospheric variation across the Sorgenfrei -Tornquist Zone, Southern Scandinavia[END_REF] found that this Zone presented a dipping steeply to the north -east, as a conclusion of their Pwave travel time residuals. Their results are then in contradiction with ours and the ones of [START_REF] Arlitt | P-wave velocity structure of the lithosphere -asthenosphere system across the TESZ in Denmark[END_REF]. However, we clearly find the same structure as this latter by using very different and independent methods applied to different phases of the signals recorded by the TOR1 experiment. As expected, the relative variations in velocities at the minimum in the asthenosphere are higher as we analyze shear-wave velocities, which are more sensitive to the partial melt. Accepting that the thickness and velocity models are now well constrained across the Sorgenfrei -Tornquist Zone, it becomes important to better understand the mechanisms that led to the juxtaposition of such different lithospheric structures.

Fig. 2 .

 2 Fig. 2. Geographical distribution of the 99 events (grey dots) used in this study. The polar representation is centered in Denmark (55jN-12jE).

Fig. 4 .

 4 Fig. 4. Dispersion curves measured for the Rayleigh wave using the D1 and D2 arrays in Denmark (a and b) and the G1 and G2 arrays in Germany (c and d). The grey area is the confidence interval that we used in the inversion for determining lithospheric models. The black dots show the average value of the measured phase velocity.

Fig. 6 .

 6 Fig. 6. Dispersion curves measured for Rayleigh waves along long profiles in Germany (a), Denmark (b) and Sweden (c). These three dispersion curves are superimposed on graph (d). The grey area is the confidence interval that we used in the inversion for determining lithospheric models. The black dots show the average values of the measured phase velocity.

Fig. 8 .

 8 Fig. 8. Lithospheric models determined by inversion of the Rayleigh wave dispersion curves under Germany (a), Denmark (b) and Sweden (c).

Fig. 9 .

 9 Fig. 9. Model of the lithosphere beneath the TOR1 profile, from Germany to Sweden through Denmark, based on the analysis of Rayleigh waves presented in the text. STZ = Sorgenfrei -Tornquist Zone. TEF = Trans-European Fault.

Table 1

 1 Lithospheric thickness (Z) in km and shear-wave velocities in the upper mantle (V S0 ) and at the minimum in the asthenosphere (V S1 ), in km/s, as determined by the inversion of Rayleigh wave dispersion curve See text for details. Bold letters are for long aperture profiles located in the three different countries (Sweden, Denmark and Germany) and the others are the arrays (D1, D2, G1 and G2).

	Area	ZV S0	V S1
	S	>200	4.66 F 0.03
	D1	>200	4.57 F 0.04
	D2	120 F 20	4.60 F 0.04	4.45 F 0.09
	D	120 F 20	4.57 F 0.06	4.43 F 0.08
	G1	50 F 10	4.43 F 0.06	4.39 F 0.05
	G2	75 F 25	4.57 F 0.08	4.36 F 0.09
	G	100 F 20	4.54 F 0.03	4.46 F 0.08

Acknowledgements NC thanks S. Gregersen for giving her the opportunuity to go to Copenhagen and to participate in the fieldwork, and also for fruitful discussions and comments of this work. The authors are indebted to A. Paul for careful commenting a preliminary version of the manuscript, and to M. Grad and an anonymous reviewer for constructive reviews. NC was supported by a contract between the Centre National de la Recherche Scientifique and the Laboratoire de De ´tection et de Ge ´ophysique, No. 72B 087/00. This is a EUROPROBE publication.


David Dureisseix

An introduction to dimensional analysis

Goals of dimensional analysis

Dimensional analysis (DA) is a wide-spread tool in fluid mechanics, but less used for structural engineering. Nevertheless, it has been implicitly present for a long time in certain applications. One example is stress concentration graphs [START_REF] Faurie | Guide du dessinateur, les concentrations de contraintes[END_REF], another the design of small scale models (for experimental quantitative studies) similitude.

Since these notes were initially part of a course on 'friction and lubrication', the mechanical components that are studied, belong to the theme of fluid-structure interaction. As such, some applications of dimensional analysis for fluids mechanics might transpose to solids mechanics. Nevertheless, not so much applications concerning the 'friction and lubrication' theme are developed in these notes, since dimensional analysis application range is wider than this particular course.

Apart for its instructional purpose on physical quantities and the system of units [START_REF] Debacq | Unités, dimensions et nombres adimensionnels[END_REF][START_REF] Sonin | The physical basis of dimensional analysis[END_REF], DA enables one to check the homogeneity of a formula. DA is often used to make physical equations dimensionless and to focus on dimensionless characteristic numbers that are useful for detecting the physical regime of the studied phenomena (Reynolds number, Mach number...). Such an analysis may eventually lead to a simplification of the model, a reduction of the number of parameters, an extension of the domain of application and the use of small scale models.

Physical quantities and their units

A physical quantity (a length, a pressure, a temperature...) is usually a compound of two parts: (i) a value (i.e. a number), and (ii) a unit which is characteristic of the kind of physical quantity that is looked at. We usually indicate that a symbol chosen to represent this quantity is equal to the value times the unit, e.g. a length is typically L = 2.5 km Such quantities are sometimes called denominate numbers.

Here we wish to point out that this is indeed a true multiplication between the value and the unit. Changing the unit does change the value but not the physical quantity: L = 2.5 km = 2 500 m = 2.5 × 10 9 µm Moreover, we can divide by a unit to extract a value:

L/m = 2 500
(it is a true division of a physical quantity by a unit). Note that this is the recommended notation for labeling the axis in a graph: values as tics, or table entries, see appendix B.1). Additionally, we can imagine dividing a physical quantity by a number:

L/2 500 = m = 1 m

To convince yourself, you can also have a look on this slot of the Khan Academy. For the purpose of good communication, we expect that all the people working together on a project understand the same physical quantities, i.e. they are aware of the units (and their abbreviations) used by the others. To avoid misunderstanding, the standardization imposes rules to denote the physical quantities and their units1 . A milestone in this standardization procedure is the 11th general conference on weights and measures, held in 1960: This is the first advent of the International System of units (SI).

Clearly, there are still other units in use, for instance the English units, the Imperial units... that are now supposed to be superseded by the SI2 . Other, even less regulated units, are linked to various professions; try to guess which professions use these somewhat strange units in table 1, after this link.

A bit of History: Definition of the meter

The first definition dates back to march 26, 1791: 1 m = 1/10 000 000 × half the length of the earth meridian. 16 standard meters (mètres-étalons) were made; only 4 remain today 3 . The idea was therefore: The value of the length of something measured in meters, is the number of times the standard length fits alongside. For instance, if L 0 was the standard meter, then L/L 0 = L/m = 2 500 (one can align 2 500 times the standard length to cover the measured distance). Note that if the standard models where not 'exactly' equal, or if their length varies in time due to temperature, hygrometry... one is running into trouble. So, with the advent of more and more precise measurements, this 'definition' itself was not accurate enough. Nowadays, the meter is defined as the length of the path traveled by light in vacuum during a time interval of 1/299 792 458 of a second [START_REF] Dursapt | Appelez-moi mètre[END_REF] (this definition may change again in the future... see appendix A.

A simple example of computing on values and on units

Consider the case where you have to evaluate the velocity of a peripheral point on a rotating solid. The angular velocity of the solid is ω and the distance of Figure 1: Angle measurement the considered point to the axis of rotation is R; you probably already know that the particular velocity is V = Rω. Now we need a numerical application: Say that the radius is R = 10 cm, and the angular velocity is ω = 1 500 rpm; what is the velocity?

Angular measurement

First, have a look at the unit of angular velocity. rpm stands for 'revolution per minute'; it is a classical unit, but not as standard as the french counterpart, for which tr/min stands for 'tour par minute'4 .

'tour' is actually a unit for measuring angles. This angle measurement is our first special case: The corresponding basic SI unit is the radian (abbreviated with rad5 ) and is defined as follows. Consider a circle of radius r and center O with two points A and B defining the angle θ, figure 1. The measure of this angle in radians is defined as the ratio of the arc length ⌢ AB over the radius: θ = ⌢ AB/r. Therefore, the radian is a unit which is... nothing (a length divided by a length) 6 . Hence the following is valid:

θ = π rad = π
nevertheless, the advice is: Keep the symbol rad to keep in memory that this is an angle! Other angle measures (not SI) are: the 'tour' (tr and the 'grad' (gon), 1 tr = 2π rad = 400 gon Note also that the second of arc ′′ and the minute of arc ′ are measures of angle, not of time:

1 ′ = (1/60) tr, 1 ′′ = (1/60) ′
The basic unit for the rotation rate is therefore rad/s. This is different from a frequency (the number of cycles per time unit) though they have formally the same unit: The hertz7 (abbreviated as Hz) is better used for the frequency, 1 Hz = 1 s -1

Using your favorite pocket calculator

Lets go back to the numerical application for the velocity

V = Rω
To obtain V , simply use your pocket calculator. Note that, up to now, this kind of computer only computes using values, not yet on physical quantities (the units are not taken into account). So we can use it as: enter the value for the radius [START_REF] Sonin | The physical basis of dimensional analysis[END_REF], multiply by the value for the rotation rate (1 500), ask for the result which is hopefully 15 000.

The problem is now: What is the associated unit?

The answer is: The unit of the result is the same computing sequence applied to the units of the input. Since we did a multiplication, the correct answer should be:

V = (10 × 1 500)(cm × tr/min) = 15 000 cm • tr/min Though correct, this result is quite cumbersome. Better is to use a more standard unit such as m/s. To change the units, perform the following calculations: cm = 10 -2 m, tr = 2π rad, min = 60 s so V = 15 000 cm • tr/min = 15 000 × (10 -2 m) × (2π rad)/(60 s) = 5π m/s

Additional remarks and (funny) special cases

Note that if additions have to be made, only quantities with the same physical meaning can be added; the following development is therefore correct:

1 m + 10 cm = (1 + 0.1) m = 1.1 m

If you consider analytical functions such as sin, log, exp... their argument should be without unit. For angles, the rad can be perfectly used in this way! Some units are compounds of a base unit with fractional exponents. This is perfectly consistent with the previous use. For instance, the crack propagation in fracture mechanics uses the stress intensity factor (in crack mode I), defined as

K I = 1.12σ √ πa/Φ
where σ is the tensile nominal stress, a is the crack length, and Φ is a coefficient depending on the crack shape. You can therefore check that the unit of K I is

[K I ] = Nm -3/2
The crack growth itself is often modeled with a Paris law of the form

da/dN = C[∆K I ] m
where N is the number of cycles, and ∆K I a variation of K I during a cycle. m is a material parameter that should have no unit, and C is a second material parameter; you can check that the unit of C is

[C] = m(Nm) 3m/2
(it depends itself on the value of m!).

To avoid the use of many zeros before the comma (for huge quantities) or after it (for small quantities), some prefixes are standardized. They are merely multiplicative values without unit. Apart the classical deca (1 da = 10), hecto (1 h = 100), deci (1 d = 0.1), centi (1 c = 0.01), the other standard ones are defined for each change of amplitude of 3 orders of magnitude, sometimes with funny names.

Concerning multiplicative factors, you can also think about the percentages: Percent merely means one over one hundred. Therefore one simply has 5% = 0.05, 4 = 0.004...

Finally, concerning the computation on values as well as on units, the special case where the value is zero can be mentioned. The zero property states that anything multiplied with zero is zero, so we can write:

V = 0 m/s = 0
To be sure that this rules applies, care must be taken for the temperature measurements... see appendix C .

3 Physical laws do not depend on a particular system of units even if you don't now what exactly the laws are... (this is a consequence of the more fundamental statement that all physical laws can be represented in a form equally valid for all observers). What is a physical law? It is a relationship between physical quantities (which are products of a value and a unit). For the previous example of the velocity, it can be expressed as

V = f (R, ω)
where f is a function of two parameters, this indicates the relation between the physical quantities V , R, and ω. Indeed, the same function could be applied to the values alone, and to the units alone, as exemplified before. Therefore, the physical law f does not depend on the choice of the system of units, provided it is consistent: Whatever this choice is for R and ω, the unit of V can be deduced.

The SI base units

The SI is first based on a set of independent base units. What is the number of independent base units? The smallest possible. This means that each time a physical law is identified and modeled, it is a relationship between different physical units that are therefore not independent. For instance, the newton is a unit for a force, but one knows that gravity produces a force F from a mass M and an acceleration g: F = M g is a physical law. The same function applies to the units, so

N = kg • m/s 2
which can be viewed as the definition of N. N is therefore a derived unit (not a base one) built on kg, m and s. Today, the number of all the base units is 7 8 . A useful notation, though not so standard, consists in using brackets to denote the extraction of the unit in the SI system from a physical quantity 9 . For instance,

[V ] = m/s, [F ] = N = kg•m•s -2 . Therefore the value of a physical quantity in the SI unit is for instance V /[V ] or F/[F ].
Now consider the initial physical law V = f (R, ω). It is valid for the units in a consistent system of units, like SI. Therefore, one can write:

[V ] = f ([R], [ω]),
and it is also valid for the values in a consistent system of units, i.e.

V /[V ] = f (R/[R], ω/[ω]
). This last expression relates dimensionless quantities!

Special choice of base units

For deriving a dimensional analysis, the Vashy-Buckingham Πs theorem is famous and the general approach for rigorous derivations. A simpler approach which is used herein is the following: Since a physical law is independent of the system of units, one can use a dedicated system of units. In the same spirit as for the artifacts, the 'mètres-étalons' may be chosen in the arguments of the physical function law itself. Indeed, the question of the values for the various quantities is therefore how many 'étalons' are there in the quantity? Note that some fundamental requirements are:

(i) the output is independent of the inputs (think of a test rig: To design and operate it, you need to specify a certain number of input parameters, such as lengths, velocities... and the measured outputs are the results of the test; for a spring you can prescribe the displacement and measure the force, or conversely, but not prescribe both), (ii) the inputs chosen as base units should be independent, i.e. they should provide all the units required for the physical problem that is studied. Their number may depend on the overall physics that is studied; for instance, if one studies geometry only, there is only one unit required: the one used for length. If one deals with kinematics, there should be two units to be able to measure length and time. If forces are also involved, (dynamics), there should be three units that allow to measure length, time and force (or mass, since N = kg•m•s -2 ).

Lets use the example of our physical law V = f (R, ω). Clearly, we need something for the lengths, as well as for the time. Two basic units are required, so we can try R and ω. Now we wish to write the same physical law for the values of each involved quantity: How many R's in R? R/R = 1, so 1 is the value of R in this new system of units; how many ω's in ω? ω/ω = 1, so 1 is also the value of ω in this new system of units. Finally, what is the value of V ? Here we can see that the choice of the two base units is enough to describe the physics: The value of V is V /(Rω) (justified by [Rω] = m/s). The physical law for the values in this new system of units is therefore:

V Rω = f Å R R , ω ω ã = f (1, 1) = constant
Without information on the detailed model for the physical law, we can nevertheless conclude that it reads

V = constant × Rω
(which is indeed consistent with the detailed result that found constant = 1). This case is a little bit special and we can consider a more generic case: If we consider that the solid with a rotational movement is very flexible, the centrifugal forces may change the form of the solid, and so, the velocity that is studied. Determining (or choosing) the input parameters is a difficult task, it relies on the expertise of the modeler. Here, we can consider that the stiffness of the material E (Young modulus) is an important parameter, as well as the Poisson coefficient ν for the elastic behavior, and the density ρ. The physical law is therefore

V = f (R, ω, E, ν, ρ)
These parameters are usually belonging to 3 possible families: (i) the geometrical parameters, (ii) the load parameters, (iii) the material parameters. With this new physics, 3 base units are required. Lets try to choose R, ω and E. You can check that the same framework as before leads to

V Rω = f 1, 1, 1, ν, ρ ER -2 ω -2 = f ν, ρ ER -2 ω -2 (1)
The number of arguments is reduced from 5 to only 210 ! The dimensional analysis actually does not introduce additional information, but rather condenses the information. Nothing more can be said about the expression (1) up to this point.

To particularize the unknown function f some more modeling, or assumptions, or experiments have to be made. The initial choice of important parameters in the description of the physical law may lead to different results. Note that for this example only one geometric parameter, R, was in the list of arguments. This means that the final result is only valid for a class of cases where only the size of the solid (given with the sole parameter R) can change, and not its shape. On the other hand, if the complete shape was to be taken into account, all the (independent) geometric parameters defining the geometry (think of a fully parameterized geometry with a CAD tool) should have been arguments of function f , but still only 3 parameters could have been gotten rid of.

Does the result depend on your initial special choice?

Consider again the physical law V = f (R, ω, E, ν, ρ). But try now as base units R, E and ρ. You can check again that this is a valid choice and that it leads to:

V E/ρ = f Ç 1, ω R -1 E/ρ , 1, ν, 1 å = ḡ Ç ω R -1 E/ρ , ν å (2)
Still 2 arguments, but not really the same expression as before (1)... One or the other expression may be more or less interesting for the user, but are they equivalent?

V = Rω × f ν, ρ ER -2 ω -2 = » E/ρ × ḡ Ç ω R -1 E/ρ , ν å ?
To simplify this expression, lets denote x = (R 2 ω 2 )/(E/ρ) and y = 1/ √ x; this can be re-casted as: Do we get

f (ν, x) = 1 √ x × ḡ( 1 √ x , ν) or ḡ(y, ν) = 1 y × f (ν, 1 y 2 ) ?
Now do not forget that f was an unknown function; so is ḡ. Therefore, we only see now what is the relation between these two unknown functions... If we know one, we now know the other one. So all is perfectly consistent.

Example of stress concentration factors (SCF)

Consider for instance the case of a cylindrical beam in traction (P is the traction force) with a change in diameter from D to d, and with a radius at the junction r, figure 2. You probably know [START_REF] Lemaitre | Résistance mécanique des solides. Matériaux et structures[END_REF] that the maximum stress in the material is reached near the diameter change, and that its value is larger than that obtained in the case of a smooth beam, i.e. a beam without diameter change, and with the smallest diameter d:

σ max > σ nom = P πd 2 /4
The question is: How to obtain this maximum stress needed for designing the beam? One may rely on experiments, simulations... but in order to build a compendium of results available to rapidly estimate the maximum stress for various geometries, materials, forces... many tests or computations will be required. To decrease this effort, one can try dimensional analysis.

The maximum stress is obtained with a certain physical law, once we decide what are the input parameters. We may consider that geometry is given with d, D, r but not the length! (we consider only diameter changes and other 'defects' far away one to the other, so that they do not interfere). Note that we suspect herein that r is an influential parameter (if it is the case, and if we did not use it as an entry, we would make a modeling mistake...) We also did not add to the inputs the length t = (D -d)/2 since it is not independent of the other ones.

Concerning the material parameters, for an linear isotropic elastic homogeneous material (an additional assumption), we could use Young modulus E and Poisson coefficient ν. We may suspect that E has no influence (at least, it has none on σ nom )... to be assumed with the user knowledge, to be checked by experiments or simulations... A usual assumption in this case is also that for classical range of the values of ν, it has only a small influence on the result.

Finally, concerning the load, indeed P has an influence.

Choosing the important parameters is one of the harder tasks. For now on, we consider that σ max = f (d, D, r, P )

The second stage is to choose the base units among the inputs. For instance: d for length, P for force. Are these independent? [d] = m, [P ] = N = kg•m•s -2 , so indeed they are. Are they sufficient to describe the physics of our problem? Let us try: [σ max ] = Pa = N/m 2 = [P/d 2 ], so the physical law can be rewritten as

σ max P d -2 = f Å 1, D d , r d , 1 ã = f Å D d , r d ã (3)
From 4 parameters, only 2 are useful! This can be depicted on a single abacus with σ max /(P d -2 ) on the vertical axis, and D/d and r/d as abscissa and parameter.

Let us look at a classical SCF chart, figure 2 (or see an other version here). This is not exactly what was expected... K t is plotted as a function of d/D and r/t. But wait: The SCF is defined as

K t = σ max /σ nom = σ max /(P × 4d -2 /π)
this is the dimensionless stress in equation ( 3) with a multiplicative constant π/4; d/D is the inverse of D/d and r/t = r/d × 2/(D/d -1). Therefore, we indeed get

K t = ḡ Å d D , r t ã = π 4 f Å D d , r d ã
Finally note that r has a tremendous influence on the maximum stress. Forgetting it at the beginning would have lead to unreliable results. We can also note that with the first dimensionless result, the maximum stress is proportional to the load P (linear elastic regime only! see genuine assumptions).

4 Some examples for practicing dimensional analysis: Train yourself

Bone is a porous media

A porous media is the mixture of a fluid and a solid: There is a fluid that can migrate inside the solid. Bone is a porous media11 and some consider this phenomena as a way to address the question of bone remodeling. The following example concerns a mice ulna [START_REF] Robling | Improved bone structure and strength after long-term mechanical loading is greatest if loading is separated into short bouts[END_REF]. A simplified (1D) case leads to the analytical pore pressure evolution equation [START_REF] Malachanne | Numerical model of bone remodeling sensitive to loading frequency through a poroelastic behavior and internal fluid movements[END_REF]:

∂ 2 p ∂x 2 = τ L 2 ṗ + α E σ
where p is the pore pressure (the pressure of the fluid inside the solid),

x ∈ [0, L] is the coordinate, τ = L 2 /H(1/Q + α 2 /E) is the characteristic time, L = 2 cm is the characteristic length, α = 0.78 is a coupling coefficient (no dimension), E = 15
GPa is Young modulus, σ is the external mechanical load (Pa), Q = 15 GPa is the so-called compressibility modulus or Biot modulus, and H = 1.1 × 10 -13 m 4 /N/s the intrinsic permeability. The problem is not well conditioned if SI units are used (see the difference of values for E or Q and H...) hence leading to bad numerical results. The question is therefore: Find a new system of units (a length, a mass, a time) Figure 2: A classical SCF chart, after [START_REF] Faurie | Guide du dessinateur, les concentrations de contraintes[END_REF], with permission such that the following quantities have a value 1: Young modulus E, intrinsic permeability H, characteristic time τ .

For checking purpose, I found a characteristic time (for the pressure accommodation) of τ ≈ 0.39 s, a length ≈ 0.025 m, a mass ≈ 5.8 × 10 7 kg.

Porous ceramics is a porous media (!)

Porous ceramic materials are sometimes used as filter for hot fluids (adding the thermal behavior to the porous behavior). For a simplified (1D) case, the temperature evolution is governed by a (tricky) advection-diffusion equation [START_REF] Néron | A computational strategy for thermoporoelastic structures with a time-space interface coupling[END_REF]:

∂ 2 θ ∂x 2 = C d k θ - ρ F c F k H P d L d ∂θ ∂x - 1 -3α F T 0 k H P 2 d L 2
d
where θ is the temperature, x ∈ [0, L d ] the coordinate. Using T 0 as a reference temperature (dimensionless temperature is θ/T 0 ), L d as a reference length (dimensionless coordinate is x/L d ), P d as a reference pressure, the dimensionless version of the previous equation is

∂ 2 (θ/T 0 ) ∂(x/L d ) 2 + P e ï ∂(θ/T 0 ) ∂(x/L d ) - ∂(θ/T 0 ) ∂(t/τ T ) ò = -B r (4)
These models involve many parameters (this is the drawback of multiphysics coupled problems!) that are: P e is Péclet number (it quantifies the order of magnitude of the heat convectively transported by the fluid compared to the heat supplied by diffusion). B r is Brinkman number (it quantifies the order of magnitude of the heat source due to the viscous dissipation compared to the heat supplied by conduction). These are both dimensionless numbers.

The question is: Give the expression and value of the reference advection time τT in the simpler dimensionless equation (4).

For checking purpose, I found: P e = 445, B r = 37 and τT = 136 s.

For biomechanics too

Maybe you have seen a TV series named 'on n'est pas que des cobayes' on channel 'France 5'. During one episode, on march 2012, they wonder who is stronger, man or ant. Let us study the same question, with an additional precision: If they were of the same size, who would be stronger? First, collect data:

• Up to now, the strongest man on earth may well be Hossein Reza Zadeh (Iran). He is a weighlifter and holds the world record for clean & jerk, super heavyweight class (over 105 kg, more precisely M = 152 kg for H = 1.86 m) with 263.5 kg lift at the 2004 Olympics Games in Athens (25/08/2004), so he can lift approx. 1.7 times his own weight.

• The strongest insect on earth may well be the Formica Rufa (red ant) that can lift approx. up to 60 times its own weight for a size of approx. h = 9 mm.

So, who's the strongest? If we define a performance index p as being the ratio of the force needed to lift, to the mass, p = F/M , we can use a crude dimensional analysis to get orders of magnitude. Consider that mass is a function of size H and mean density ρ: M = f (H, ρ), and that the lift force is a function of the size H and of the strength limit R p for the tendons, muscles or bones (depending which is the first to break), F = g(H, R p ).

For the first physical law, use as base units H and ρ, for the second one, we may use H and R p . Try these and give an expression for the performance index p. Now you can answer the question: What happens to the performance index of the human if its size is reduced to that of an ant?

For checking purpose, I found that it is multiplied by a factor of approx. 200. Therefore, if the real Hussein can lift 1.7 times his own weight, the resized one could lift 340 times his own weight. So, who's the strongest?

As an addendum, I recently read a related article [START_REF] Lehoucq | Superman est-il une super-fourmi ?[END_REF] with the same flavor; maybe you read it too?

Drag

Maybe you know about the C x (drag) coefficient? It is related to the aerodynamic resistance to movement. Consider a car with a velocity V ; can we quantify and compare the resisting horizontal force F due to the action of the air (without wind) on the car?

The influential parameters are all that define the geometry of the car body (say lengths l i ), the velocity V , the air density ρ: F = f (l i , V, ρ). Usually, the base units (need for 3) are: the projected surface on the direction of the velocity S ('maître-couple'), the velocity V and the density ρ.

Apply the dimensional analysis to find that

F = 1 2 ρV 2 SC x
where C x is a dimensionless coefficient, called drag coefficient, function of the adimensional geometry of the vehicle. Different cars have different drag coefficients. The world record for efficiency seems to be the PAC-Car II prototype (Shell Eco-marathon in Nogaro, 2005) with SC x = 0.019 m 2 . For fuel consumption issues, note also that the needed power at velocity V is P = F V = 1 2 ρV 3 SC x so the velocity has a huge influence... Moreover, we may not be really interested in the value of the C x coefficient, but rather in the value of SC x , so beware of the advertising arguments! To convince yourself, you may compare the C x of

• the Empire state building, with a drag coefficient of 1.3 to 1.5,

• the Eiffel tower, with a larger drag coefficient of 1.8 to 2!

Certainly not the same resistance to wind... consider the values for S...

Using small scale models

When designing (sometimes unique) large-scale structures, one needs to perform experiments. Nevertheless, building a real-scale prototype is often too costly, so the question is to built a model at a reduced size, perform measurements on this model, and deduce what would have been the same measurements on the full-scale case. Changing the scale, and eventually other parameters, can be assessed with dimensional analysis.

Example

Consider for instance that one wishes to design a new passenger ship hull. One is interested in the resistant force F that needs to be overcome by the propulsion system. The full fluid-structure with a free surface problem is complicated enough to rely on experiments rather that on simulations. Since a passenger ship is often a unique product design with a large cost, as for the Queen Mary II, the prototype is the final product and experiments should be done on cheaper small scale models. This problem is somehow more intricate than the drag force for the car; the influential parameters could now be: the various lengths l i that parameterize the hull geometry (including the position of the free surface, the waterline), the gravity g (that influences the waves generated by the movement), the viscosity η and density ρ if the fluid, the velocity V . Therefore, the physical law reads F = f (V, l i , g, η, ρ)

The chosen base units could be a particular length L, velocity V and density ρ. Show that the dimensionless drag force is

F ρV 2 L 2 = f Å l i L , g L -1 V 2 , η ρV L ã
Since f is an unknown function, the force can be estimated from a measurement on a small scale model if the arguments of the function f are the same for the full scale problem and the model. The equality of the argument values are the similitude conditions. The first of these conditions is to have the same nondimensionalized geometry lengths, meaning that the scale should be preserved without distortion of the geometry. The second one is the equality of the nondimensionalized gravity (which is called the Froude number); since gravity will be the same for the real scale and small scale cases (!), the coefficient L -1 V 2 should be the same. The last one is the equality of the non-dimensionalized viscosity (which is called the Reynolds number).

Show that measuring the force as a function of the velocity on the small scale model (in a towing tank, 'bassin des carènes', see for instance this one at ECN), with subscript s (i.e. F s = h(V s ) is an experimentally plotted function), therefore provides the force as a function of the velocity on the real scale case, with subscript r:

F r = ρ r ρ s Å L r V r L s V s ã 2 × h Ç L s L r V r å
One can notice that equalling all the arguments may not be a trivial task: for instance, using the same fluid would lead to have the coefficient LV equal, but Froude said that L -1 V 2 should also be equal, which is not possible (unless scale 1 is used...) Considering other influential parameters, such as surface tension (negligible here!), would have lead to an other dimensionless condition with an non-dimensional number (the Weber number) and would have complicated the similitude conditions.

Large scale model as well...

The same tool can be used in the reverse direction: If a real-scale prototype is too small to be built at the early design stage (micro-mechanisms that would require specific manufacturing tools not available at that time of the design process), an enlarged model can be manufactured more easily, and experiments can be conducted on it. Going back to the real-scale case is the same problem as before. A A future change in SI basic units definition?

A previous release of this document states "the future is coming: the SI roadmap planed the new unit definition to take place in 2018, see this link" (the link is now broken...). Now the future is the past: the vote had taken place on friday 16, 2018, during an open session of the 26th General Conference on Weights and Measures (CGPM) at Versailles, France. The base units are not defined by objects anymore, but with values of fundamental constants of nature. The session could be followed in streaming at youtube.com/thebipm channel, especially with the video (note the historical agreement of many countries) "On the revision of the International System of Units (SI)" and Voting, and the (short) advertising/teasing "We Did It! Redefinition of SI adopted".

Finally, for having a moment of fun, you can read the article [START_REF] Berry | Manifeste pour la réhabilitation du Pavillon des Poids et Mesures. Viridis Candela[END_REF] of Gérard Berry concerning the 'Pavillon des Poids et Mesures'.

B Some not-so-clear uses and alternative solutions

B.1 Graphs and tables

You may have seen that scientists often use 2D plots or tables of values for giving some physical quantities. On a 2D-plot, figure 3, the axis are graduated with values and you would certainly find that repeating for each the same unit is cumbersome. Therefore, only values are given. But we have to specify somewhere the unit that is used! You will sometimes find the units only in the caption of the figure, or eventually in the form 'R (m)' for designing a radius R expressed in meters... and if one wishes to give the decimeter values? Will it use 'R (0.1 m)' ? The standards recommend to use the same consistent and unambiguous notation as herein: The value of R when the meter is used as a unit, is R/m. Therefore, the corresponding axes should mention R/m, R/(0.

B.2 Physical law valid for special units only

You may sometimes find some expressions that are only valid for a certain choice of units. This does not mean that they are not a physical law, but only that, for practical purpose, they are tuned for using special units. Though easy to use, they nevertheless require some practice to avoid mistakes. On the other hand; the true expression conforming to a general physical law may be more cumbersome and require some coefficients with their own units. For instance, a lathe (a machine tool) requires to tune the rotation rate N to obtain a good cutting velocity V depending on the material of the tool and on the part to be cut, to create a good cylinder of diameter d. The practical expression is N = 1 000 × V /(πd), valid only if V is given in m•s -1 , N in tr•s -1 and d in mm. The correct corresponding relation with our development is N/(tr/s) = 1 000 (V /(m/s))/(π(d/mm)), or N = 1 000 (mm × tr/s)/(π m/s) V /d = V /(d/2) that does not involve any units; therefore this is the true physical law.

B.3 Not-so-clear abbreviations for units

Some compound units may be a combination of many base unit. When the units are multiplied and divided in various arrangements, using exponents may clarify the case. For instance, a specific heat capacity is measured with an energy per unit of mass and per unit of temperature: The notation J.kg -1 .K -1 is less ambiguous than, and therefore preferred to, J/ kg.K, or J/(kg.K), or J/kg/K.

Concerning the mechanical torque, the case is delicate. Indeed a torque is a force (for which a SI unit is the newton, abbreviated as N) multiplied with a length. Therefore its unit is N•m. A first remark is: Avoid permutation of these two units, since m•N could be easily read as 'milli newton' (mN)! The second remark is that a force times a distance is also a mechanical work, for which the unit is the joule (abbreviated as J) 12 .

Formally, a torque and a work are homogeneous, but advice is: keep the symbol N•m for the torque, and J for the work to keep in mind the nature of the physical quantities that are manipulated.

C The temperature case

Care must be taken when dealing with temperatures. Indeed, the absolute temperature should be measured in kelvin (K) 13 . Other units, such as degree

  Reference temperature T 0 = 293 K Characteristic pore pressure P d = 36 MPa Characteristic length L d = 1.4 m Permeability of the porous media H = 2 × 10 -10 m 3 s/kg Thermal conductivity of the porous media k = 68 W/m/K Specific mass of the fluid ρ F = 1 000 kg/m 3 Specific heat of the fluid c F = 4 182 J/kg/K Heat capacity of the porous media C d = 2.1 × 10 6 J/K/m 3 Thermal expansion coefficient of the fluid α F = 2.6 × 10 -4 K -1

Figure 3 :

 3 Figure 3: Two plot presentations (the one on the right is standardized and less ambiguous)

Table 1 :

 1 Some strange units (still in use)

	name	abbrev. usage	remarks
	knot	kt	unit for speed,	1 knot =
			nautical	1 nautical mile per hour
	horsepower hp	unit of power	1 hp = 746 W
	acre	ac	unit of area	1 acre = 4840 square yards
	barrel		unit for volume, now	1 barrel = 42 US gallons
			much useful for oil, US
	barrel		unit for volume,	1 barrel = 36 gallons, or
			for beer	4 firkins, or 288 pints
	slug	SL	unit for mass, old	mass which provides a force
			British system	of 1 pound when subjected
				to an acceleration of 1 ft/s 2

Table 2 :

 2 1 m) or R/dm, etc. Same thing for different values given in a table, see table 2. Two table presentations (the one on the right is standardized and less ambiguous)

	velocity (×10 m/s) error (%)	velocity / (10 m/s) error / %
	1.0	10.1	1.0	10.1
	2.1	15.0	2.1	15.0
	4.2	18.1	4.2	18.1
	6.1	22.5	6.1	22.5

try to browse this one, this other one, or even this one

see this link to be aware that this is not yet the case

have a look at this, to locate this one

note that the duration unit 'minute' is abbreviated as min and not mn!

and not rd which is the 'rad': An old unit for radiation quantity, now superseded by the 'gray'

There is also a measure for the solid angle which is the steradian, defined as a surface to surface ratio

It seems that the units, even when derived from a name, are not expected to exhibit an uppercase letter... except when this is a name, such as 'degree Celsius'... but the symbol does... a bit strange, however.

see this link, or this one

in particular, we get for an angle [θ] = rad = 1

note also that ν has not been changed. Indeed it was initially already a dimensionless coefficient, that has the same value on every system of units!

Not convinced? See this image.

Thanks to Aurélien Pommel, student in mechanical engineering at INSA Lyon for debugging this...

 13 Note that no °symbol is used with K.

D For L A T E X or pdfL A T E X fans

Consider using the recent siunitx package. It has been used to typeset this document.