G Morlino

P Blasi

E Amato

Keywords:

The supernova paradigm for the origin of galactic cosmic rays can be tested using multifrequency observations of both non-thermal and thermal emission from supernova remnants. The smoking gun of hadronic acceleration in these sources can, however, only be provided by the detection of a high energy neutrino signal. Here we apply the theory of non-linear particle acceleration at supernova shocks to the case of the supernova remnant RX J1713.7-3946, which is becoming the stereotypical example of a possible hadronic accelerator after the detection of high energy gamma rays by the HESS telescope. Our aim is twofold: on one hand we want to address the uncertainties in the discrimination between a hadronic and a leptonic interpretation of the gamma ray emission, mainly related to the possibility of a statistical uncertainty in the energy determination of the gamma ray photons in the TeV region. On the other we want to stress how a km 3 neutrino telescope would break the degeneracy and provide evidence for efficient cosmic ray acceleration in RX J1713.7-3946. A 3σ evidence would require about two years of observation.

Introduction

Supernova remnants (SNRs) are the most plausible sources of galactic cosmic rays. The recent detection of multi-TeV gamma radiation from several SNRs makes the case stronger, especially when coupled with multifrequency observations of the same remnants. However, despite all this progress, the evidence that SNRs are indeed the main contributors to Galactic cosmic rays remains circumstantial. A smoking gun evidence of efficient acceleration of cosmic rays in these sources can only come from the detection of high energy neutrinos, resulting from the decays of charged pions within the source.

The acceleration of cosmic rays at supernova blast waves is well described by the non linear theory of diffusive shock acceleration (NLDSA) (see [28] for a review).

Email addresses: morlino@arcetri.astro.it (G. Morlino), blasi@arcetri.astro.it (P. Blasi), amato@arcetri.astro.it (E. Amato) This theory allows us to calculate the spectrum and spatial distribution of cosmic rays accelerated at a supernova shock taking into account the dynamical reaction of the accelerated particles on the shock and, in its most recent version [6,16,17], also the generation of magnetic field through streaming instability induced by the accelerated particles [13,12] and the dynamical reaction of the amplified magnetic field on the plasma. Such a dramatic improvement in the quality of our theoretical approach allows us to finally compare theory with observations and make testable predictions for future observations. One of the most important recent breakthroughs in establishing the SNR paradigm for the origin of cosmic rays has been the detection of narrow filaments of non-thermal X-ray emission in the direction of supernova remnant rims [10,9,26,25,40]. The filaments are most commonly interpreted as the result of severe synchrotron energy losses of ultra relativistic electrons which are forced to radiate a large fraction of their energy (in the form of X-rays) in a narrow region downstream of the shock. The required strengths of the magnetic field are of order ∼ 100-500µG downstream of the shock. Such large levels of magnetization might be the manifestation of magnetic field amplification in the upstream region, as due to the streaming of cosmic rays [35,13,6]. Not only the detection of large magnetic fields signals for efficient cosmic ray acceleration, but in turn large magnetic fields are needed for increasing the maximum energy of accelerated particles to the knee region [14], namely for efficient acceleration. Efficient cosmic ray acceleration and magnetic field amplification are two sides of the same coin.

On the other hand, it has been proposed that the interpretation of the narrow X-ray rims may be flawed [32]: they could in fact result from damping of magnetic field in the downstream region. The emission region would be narrow because the field disappears, not because particles lose energy effectively. The situation is currently subject of active debate: the damping would naively result in the appearance of filaments not only in X-rays, but also in the radio emission, and at present there is no evidence for such a phenomenon. In addition, the absence of magnetic field amplification (or a mitigation of the effect) would reduce the maximum energy achieved by the accelerated particles, unless the shock configuration is quasiperpendicular [23]. An additional possibility has been proposed in [20]: the magnetic field could be amplified downstream (and not upstream) because of the development of corrugations on the shock surface which result in eddies in which the magnetic field winds up and gets amplified. In order for this scenario to lead to large maximum momentum of the accelerated particles, the field direction must be very inclined with respect to the shock normal in the upstream region.

A powerful diagnostic tool for particle acceleration in SNRs is represented by the gamma ray emission in the GeV-TeV energy region. The spectrum of the gamma ray emission, its extension to high energies and the shape of the cutoff all provide precious information on whether the radiation is of leptonic or hadronic origin. In [30] we discussed in detail the application of NLDSA to the case of the supernova remnant RX J1713.7-3946. If the HESS data on this remnant are taken at face value, then it is problematic to fit them in the context of a leptonic model for at least two reasons: 1) the spectral cutoff expected in leptonic scenarios leads to a gamma ray emission which falls short of the highest energy data points measured by the HESS telescope; 2) in any case, fitting both the TeV and X-ray data requires the presence of a very large background of infrared photons in the remnant.

The second point raised above should be considered as a circumstantial evidence against a leptonic model, but by itself it does not suffice to rule out this class of models. In principle the first point is rather solid, but experimental effects might weaken its significance: a statistical (and systematic) uncertainty in the energy determination affects the steep gamma ray spectrum by making it look smoother than it actually is so that convolving the theoretical prediction with an estimate of the uncertainty in the energy determination makes the leptonic spectrum look more similar to the hadronic case. We address this issue in a quantitative way here.

The hadronic interpretation appears to be more sound, but there are shortcomings in this case as well: first, the thermal X-ray flux expected from RX J1713.7-3946 is larger than the observed radiation of synchrotron origin. Second, the number of electrons needed to explain the observations is about 10 -4 of the number density of accelerated protons, at odds with the e/p ratio observed at Earth at energy ∼ 1 -10 GeV.

Both these points are however rather weak at the present time. The first is based on the assumption that electrons and protons share the same temperature downstream. This condition is hardly achievable and in fact one can easily argue that the temperature of electrons should be much smaller than that of protons. On the other hand, Coulomb scattering might be sufficient to raise the electron temperature to a level large enough to excite emission lines of heavy elements [31]. The second point is equally weak in that electrons might be accelerated at different stages of the SNR. Moreover, recent data from PAMELA [1] and ATIC [18] suggest that a substantial contribution to the observed electron spectrum at Earth might come from sources other than SNRs [22,37].

The safest way of proving or rejecting acceleration of hadrons in RX J1713.7-3946, as well as in other remnants, is to search for neutrinos produced in the decays of charged pions. In this paper we apply the NLDSA model developed by [5,6], and previously used to describe the multifrequency spectrum of RX J1713.7-3946 [30], in order to calculate the expected neutrino flux from this remnant.

Previous attempts at estimating the neutrino flux from SNRs have been typically based on phenomenological approaches, building on the assumption of a power law approximation for the gamma ray spectrum and simple scaling relations between the gamma ray and neutrino spectra.

In [8] the authors estimated a flux of 40 neutrino induced muons (and antimuons) per km 2 per year from RX J1713.7-3946, using a power law for the γ-ray spectrum f γ (E) ∝ E -2 , based on CANGAROO observations; the maximum neutrino energy was assumed to be ∼ 10 TeV. Neutrino oscillations, absorption and location of the detector were not taken into account. Such effects were included by [19], where f γ (E) ∝ E -2.2 was used, the power law index being inferred from HESS data (no cut-off energy was assumed in this approach). The authors obtain a flux of neutrino induced muons N µ+ μ ≃ 10 km -2 yr -1 . Again based on HESS data, [41] used two different parametrizations for the γ-ray spectra, a power law plus an exponential cutoff at E γ,max = 12 TeV and a broken power law (with a knee at 6.7 TeV), leading to N µ+ μ = 4.8 and 5.4 km -2 yr -1 respectively. An attempt to extract the proton spectrum from the HESS data on the gamma ray flux (assumed to be of hadronic origin), and compute the neutrino flux based on that, was done in [39].

In the present paper we carry out the calculations by using our model of NLDSA which provides a self-consistent description of the acceleration of cosmic rays in the remnant. This leads to a shape of the gamma ray spectrum which is not a simple power law, due to the non linear effects induced by the dynamical reaction of the accelerated particles and the amplified magnetic field. At the same time we also obtain self-consistently the neutrino spectrum, and we use it to derive the expected number of events in a km 3 neutrino telescope.

The paper is organized as follows: in § 2 we summarize the technical aspects of NLDSA and its application to particle acceleration in supernova remnants. We also describe the calculations of the non thermal radiation from RX J1713.7-3946, with special attention for the gamma ray emission in both the hadronic and leptonic scenario. We discuss in detail the possibility to use present and future observations to discriminate between the two, once a statistical uncertainty in the energy determination of the photon events is taken into account. We show that a leptonic scenario convolved with a gaussian distribution of the photon energies with δE/E ∼ 30% or larger leads to the impossibility to distinguish the leptonic prediction from the hadronic one, at least using the HESS data on RX J1713.7-3946. The implications for future gamma ray telescopes are also briefly discussed. On these premises it is very important to aim at the detection of the associated neutrino signal, whose intensity is calculated in § 3, where we also estimate the neutrino induced muon signal in a km 3 neutrino detector. We conclude in § 4.

NLDSA and the non thermal emission of RX J1713.7-3946

Spectrum of accelerated protons and electrons

In NLDSA theory, the overall shock structure and the outcome of the particle acceleration process are inextricably linked. When acceleration is efficient, the pressure of accelerated particles affects the shock dynamics, leading to the formation of a precursor, namely a region where the fluid velocity progressively decreases while approaching the shock from far upstream. At the same time the streaming of accelerated particles is responsible for the instabilities that lead to magnetic field amplification. In turn, the fluid profile in the precursor and the amplified, turbulent magnetic field, with the scattering it provides (and possibly the induced energy losses), determine the efficiency of particle acceleration and the resulting spectrum, including its high energy cutoff.

The shock structure and the accelerated particle spectrum are computed as in [30]: the basic structure of the calculation is the same proposed in [5] and [6], but with a crucial new aspect taken into account, namely the dynamical reaction of the self-generated magnetic field, which is included following the treatment of [16] and [17]. This means that the conservation equations at the shock and in the precursor are modified so as to include the magnetic contribution. The compression factor at the subshock, R sub , and the total compression factor, R tot , are deeply affected by this change, resulting in a decrease of the compression ratio in the precursor, R tot /R sub , as soon as the amplified magnetic field contributes a pressure comparable to that of the thermal gas upstream. This smoothening of the precursor reflects in spectra of accelerated particles which are closer to power laws, though the concavity typical of NLDSA remains visible [17].

The normalization of the proton spectrum is an output of our non linear calculation, once a recipe for injection has been established. Following [15], particles are injected immediately downstream of the subshock. The fraction of particles crossing the shock surface which are injected in the acceleration process, η inj , can be written as

η inj = 4/ 3 √ π (R sub -1) ξ 3 e -ξ 2 . ( 1
)
Here ξ ∼ 2 -4 is defined by the relation p inj = ξ p th,2 , where p th,2 is the momentum of thermal particles downstream. ξ parametrizes the poorly known microphysics of the injection process, but p th,2 is an output of the problem: as a result, the injection efficiency is affected by the dynamical reaction exerted by the accelerated particles and by the amplified magnetic field. Finally, the maximum momentum of the accelerated particles is determined following [14] for the computation of the acceleration time in the presence of a precursor.

For protons we use as a prescription the equality between the acceleration time and the age of the SNR:

t acc (p p,max ) = t S NR . (2)
For electrons, energy losses can be important. Their maximum momentum p e,max is determined by equating the acceleration time with the minimum between the time for energy losses and the age of the remnant. The loss time of electrons over a cycle of shock crossing needs to be weighed by the residence times, t r , upstream and downstream, so that the condition for the maximum momentum, in the loss dominated case, can be written as:

t acc (p) = t r,1 (p) + t r,2 (p) t r,1 (p) τ l,1 (B 1 ,p) + t r,2 (p) τ l,2 (B 2 ,p) (3)
where τ l denotes the loss time, and the indexes "1" and "2" refer to quantities measured upstream and downstream respectively. The residence times in the context of the non linear theory of particle acceleration can be written explicitely (from Eqs. ( 25) and ( 26) of [14]). Eq. ( 3) must be solved numerically for p e,max , contrary to the case of acceleration in the test particle regime. However an approximate analytical solution, valid when only synchrotron losses are important, was also proposed in [30].

As to the electron spectrum at the shock, f e,0 (p), this is easy to calculate for p ≪ p e,max . In fact, at a given momentum p, the slope of the electron and proton spectrum is the same, if one assumes that both species experience the same diffusion coefficient. What is unconstrained a priori is the relative normalization of the two spectra, K ep , which can only be obtained by fitting the observations. This is reasonable since electrons do not exert any appreciable dynamical reaction on the shock.

The spectrum of electrons at energies around and above p e,max , namely the shape of the cutoff is harder to calculate in the context of non-linear theory. Since the spectra we find for electrons at p < p e,max are not far from being power laws with slope ∼ 4, we adopt the modification factor calculated by [44] for strong shocks in test particle regime. The resulting electron spectrum at the shock, in the loss dominated case, is: (4) What is important to notice in this expression is that the cutoff is not a simple exponential, a fact which reflects in the shape of the synchrotron spectrum radiated by the electrons, making it different from what assumed by most of the previous work on the subject. On the other hand, if the maximum momentum is indeed determined by the age of the remnant, then the cutoff shape is expected to be exponential.

f e,0 (p) = K ep f p,0 (p) 1 + 0.523 p/p e,max

Magnetic field amplification and compression

The turbulent magnetic field close to the shock can be enhanced by several physical processes. However, here, as in [30], we focus on the amplification due to resonantly excited streaming instability induced by cosmic rays accelerated at the shock. Resonant streaming instability [35] is likely responsible for most of the magnetic field amplification in SNRs after the beginning of the Sedov phase [7], while the non-resonant mode of the same instability [12] is more effective at earlier times. When the predictions of linear theory are extrapolated to the non-linear regime of field amplification (which one is forced to do for lack of a better treatment), the resulting field strengths are in agreement with the values inferred by identifying the thickness of the X-ray filaments with the synchrotron loss length of the highest energy electrons. The strength of the magnetic field at the position x upstream, δB(x), in the absence of damping, can be estimated from the saturation condition, that, for modified shocks, reads [17]:

p w (x) = U(x) -3/2 1 -U(x) 2 4 M A,0 , (5)
where p w (x) = δB(x) 2 /(8πρ 0 u 2 0 ) is the magnetic pressure normalized to the incoming momentum flux at upstream infinity, U(x) = u(x)/u 0 and M A,0 = u 0 /v A with v A the Alfvén velocity at upstream infinity, where only the background magnetic field, B 0 , assumed parallel to the shock normal, is present. Eq. ( 5) correctly describes the effect of compression in the shock precursor through the term U(x) -3/2 . For the upstream temperature that we adopt in RX J1713.7-3946 (see below), damping in the upstream region is expected to be negligible.

The magnetic field downstream of the subshock is further enhanced by compression, according to:

B 2 = R sub B 1 , (6)
where B 1 is the magnetic field immediately upstream of the subshock and we have used the fact that we are dealing with Alfvén waves, and hence turbulence perpendicular to the shock normal.

Computation of the radiation fluxes and spectra

The flux of non-thermal radiation at γ-ray photon energies is computed as in [30]. In that work we considered both possible scenarios for the origin of high energy photons in RX J1713.7-3946, namely a leptonic origin, through inverse Compton scattering (ICS) of ambient low energy photons by accelerated electrons, or a hadronic origin, from π 0 decay following nuclear collisions of relativistic protons.

In the hadronic scenario, the γ-ray flux from RX J1713.7-3946 is accompanied by a flux of neutrinos coming from the decay of charged pions that are produced in nuclear collisions together with neutral pions. We compute the fluxes and spectra in both channels using the approximated expressions for the cross sections as provided in Ref. [24].

The γ-ray and the neutrino fluxes, Φ 0 i (E i ) (i = γ, ν), produced by p-p collisions from a source located at distance d from Earth, can be expressed as follows:

Φ 0 i (E i ) = c 4πd 2
dr n(r)

∞ E i dE p f p (r, E p ) dσ i (E p , E i ) dE i .
(7) In this expression, the apex 0 is used to indicate the neutrino flux that would have been seen at the Earth in the absence of neutrino oscillations. For gamma rays the two fluxes clearly coincide. Here n is the gas (target) density in the SNR and f p (r, E p ) is the distribution function of accelerated protons at the location r at a given energy E p . In general both quantities depend on the location in the shell, but here we assume them to be constant in the region enclosed between the contact discontinuity and the forward shock and vanishing outside this region. Finally, dσ i /dE i is the inclusive differential cross section for the production of particles of type i. This quantity is usually expressed through the total inelastic p-p cross section, σ inel , and the dimensionless distribution function F i (x i , E p ) for secondaries, as

dσ i (E p , E i ) dE i = σ inel (E p ) E p F i x i , E p , (8)
where x i ≡ E i /E p is the fraction of proton energy transferred to the secondary particle. For the functions F i , which effectively enclose all the details of the hadronic processes involved, we use the analytical approximation derived in [24] on the basis of numerical simulations of p-p collisions with the publicly available code SIBYLL.

The analytical formulae provide a very good description of the flux and energy distribution of secondaries for energies above 0.1 TeV. As far as photons are concerned, F γ also includes the contribution of η meson decay, in addition to that of π 0 , with an overall accuracy of order a few %. The estimate of neutrino fluxes is slightly less accurate, because F ν only includes the decay of charged pions, while neglecting the contribution from K-mesons, and therefore leading to underestimate the neutrino flux by about 10 %. At energies lower than 100 GeV, and down to the rest energy of the π-meson, we use the extrapolated formulae provided again by [24], that should be accurate within 10 %.

The flux and spectrum of ICS photons is calculated by using the exact kernel for ICS, with the full Klein-Nishina (KN) cross section [START_REF] Rybicki | Radiative Processes In Astrophysics[END_REF]. The main target photon field contributing to emission in the high energy γ-ray band is provided by dust-processed infrared photons, described by a blackbody spectrum with temperature ∼ 20 K [START_REF] Schlegel | [END_REF]. Following the instance of the IR+Optical photon background in the interstellar medium (ISM) we assume that the ratio of the optical to infrared energy densities remains ∼ 20, while the energy density of IR light, W IR , is left as a free parameter (in the ISM, W IR ≈ 0.05 eV cm -3 ).

In order to compute the spectra of accelerated particles and the resulting emission, we need estimates for a number of environmental parameters relative to RX J1713.7-3946, namely its distance (d), age (t S NR ), expansion velocity (u 0 ), and the values of temperature (T 0 ), density (n 0 ), and magnetic field strength (B 0 ), in the surrounding medium. In addition to these, the only free parameters of the model are: ξ, entering Eq. ( 1), and K ep , defining the ratio between accelerated electrons and protons. In the ICS scenario an additional free parameter is the above mentioned W IR .

The uncertainties on the various parameters and how they affect the results of our calculations are thoroughly discussed in [30]. Here we only summarize the values that we have found to provide the best fit to the multifrequency data in both scenarios. Our adopted values of distance, age, expansion velocity and temperature are: d = 1 kpc, t S NR = 1600 yr (consistent with the historical chinese record of a supernova explosion in AD 393 [43]), u 0 = 4300 km/s, T 0 = 10 6 K. Other parameters are, in the hadronic scenario: n 0 = 0.12 cm -3 , B 0 = 2.6 µG, ξ = 3.8; in the leptonic scenario, instead: n 0 = 0.01 cm -3 , B 0 = 1.5 µG, ξ = 4.1.

The resulting values of magnetic field strengths, acceleration efficiency and maximum energy of the accelerated particles are quite different in the two cases: the hadronic scenario entails efficient acceleration, with a fraction of accelerated particles corresponding to about 10 -4 and a maximum proton energy exceeding 10 14 eV. In the leptonic case the fraction of accelerated protons drops to less than 10 -5 , corresponding to an energy conversion efficiency of about 2 %. The magnetic field downstream is of order 20 µG, to be compared with B 2 ∼ 100 µG for the first scenario (in agreement with the value inferred from the synchrotron loss length interpretation of the rim thickness). The ratio between the number density of accelerated electrons and protons at the shock turns out to be K ep ∼ 10 -4 if acceleration is efficient. On the other hand, inefficient acceleration, and the lower value of the magnetic field associated with the leptonic scenario, would favour K ep ∼ 10 -2 , in agreement with measurements of the diffuse galactic cosmic rays. However, in order to fit both the X-ray and the gamma-ray fluxes in the context of this purely leptonic scenario a local energy density of infrared radiation ∼ 24 times larger than the galactic average is required. In Fig. 1 we plot the gamma ray flux as obtained through our calculations, for both a hadronic model (thick solid line) and a leptonic one (thick dashed line) (see [30]). Taken at face value, the curves in Fig. 1 clearly show that the hadronic scenario reproduces the high energy observations much better, in particular the highest energy data points of HESS. On the other hand, the possibility to discriminate between the two models relies upon the assumption that the statistical uncertainty in the energy determination is sufficiently small. In order to address this point, we calculate the theoretical prediction for the leptonic scenario in the case in which there is a statistical uncertainty in the energy determination σ(E) = χE with χ = 0.2, 0.3, 0.4. Our results are shown again in Fig. 1 (different line-types are labelled in the figure).

The thin curves in the figure are obtained by convolving the predicted ICS flux at a given energy, φ ICS (E), with a gaussian of given width, namely :

Φ ′ ICS (E) = dE ′ Φ ICS (E ′ ) 2π σ 2 (E) exp - (E -E ′ ) 2 2σ 2 (E) . ( 9
)
The conclusion that these curves lead to is that a statistical uncertainty larger than 30% in the energy region above 10 TeV would inhibit the discrimination between the two models, based on HESS data alone. The nominal energy resolution of HESS telescope is around 15% [4], sufficient to allow for such discrimination. The main issue in this case is the data statistics: the main differences between the two scenarios, and the superior quality of the fit obtained within the hadronic one, show at the highest energy, where the statistical significance of the data points is not very high, as shown by the large error bars in Fig. 1. The role of the energy resolution is however important also in view of future gamma-ray telescopes, like CTA, whose design is currently being discussed by the scientific community.

In addition to all this, it is worth keeping in mind that the Fermi satellite is expected soon to provide another crucial bit of information to this debate, in that the level of detection (or the non detection) should clarify the issue of a leptonic or hadronic origin for the gamma ray emission from RX J1713.7-3946, independent of the shape of the high energy cutoff (see the low energy part of Fig. 1).

Neutrino signal

The computation of the neutrino flux in the absence of neutrino oscillations was carried out in the assumption of perfect isospin symmetry (Φ π 0 ≃ Φ π + ≃ Φ π -), which leads to equal neutrino and antineutrino fluxes of a given flavor. For our purposes this is a good approximation.

The neutrino flux at the Earth is related to Φ 0 ν through the oscillation probabilities:

Φ ν l (E, d) = l ′ =e,µ,τ P ll ′ (E, d) Φ 0 ν l ′ (E) . (10)
The transition probabilities, P ll ′ , are in general functions of energy and travel length, but since at the energies considered here the oscillation lengths are very short compared with the typical size of a SNR, the oscillation probability can be space averaged. The resulting flux of ν µ crossing the Earth, not taking into account absorption, is:

Φ ν µ = P µµ Φ 0 ν µ + P eµ Φ 0 ν e = 0.4 Φ 0 ν µ + 0.2 Φ 0 ν e , (11)
with an identical equation holding for antineutrinos. The errors due to the uncertainties in the oscillation parameters are negligible (∼ 5%). Since at the source ν µ and ν e are produced in a ratio {2 : 1}, the effect of oscillations translates into a flux of muon neutrinos at Earth that is ∼ 50% of that produced at the source: Φ ν µ = 0.5Φ 0 ν µ . The flux of neutrinos and antineutrinos of each flavor expected at Earth is shown in Fig. 2, and compared with the flux of gamma rays (solid line). The flux of neutrinos has to be compared with the background whose main contribution comes from atmospheric neutrinos. The shaded region shown in Fig. 2 refers to the theoretical prediction for the atmospheric neutrino flux as we explain below.

Following the estimates of [42,21,2] the atmospheric neutrino flux above 1 TeV can be approximated as:

Φ atm ν µ +ν µ (E ν ) ≃ 4.6 × 10 -8 E ν 1 TeV -3.7
TeV -1 cm -2 s -1 sr -1 (12) with an uncertainty of ∼ 40% due to the experimental error on the primary CR spectrum and composition and on theoretical models of hadronic interactions. Notice that the background flux also depends on zenith angle due to the different thickness of atmosphere to be crossed by cosmic rays coming from different directions. This dependence is included in the thickness of the shaded region by averaging on all possible neutrinos' incoming directions (see [11] for a detailed description of the mean uncertainties in the neutrino flux determination). Notice that for atmospheric neutrinos we neglect the contribution of oscillations, which is relevant only for energies < ∼ 10 GeV [27, §6.1].

With respect to Eq. ( 12) the atmospheric background shown in Fig. 2 is rescaled to a solid angle corresponding to a cone of semi-aperture 0.5 degrees. The motivation for choosing this value of the angle is twofold: it represents a reasonable estimate for the angular resolution of a neutrino telescope at these energies, and it also corresponds to approximately the angular size of the shell of RX J1713.7-3946. A smaller value of the aperture angle would imply a better chance to detect the signal than estimated below.

In the following we specialize our predictions to the case of a km 3 tions of ν µ and νµ in the matter just below the detector. For the computation of the production rates of µ and μ, we follow the method of Ref. [19].

Since neutrinos can only be detected when the source is below the horizon of the detector, we introduce an average live-time of the source, in the form of a parameter f liv , representing the fraction of time during which this condition is satisfied. For a neutrino telescope as ANTARES and RX J1713.7-3946 as the source, one can estimate f liv ≈ 78% [19]. On the other hand, absorption of neutrinos while crossing the Earth leads to an energy dependent reduction of the detected flux. In fact the Earth becomes opaque to neutrinos at energies E ν > ∼ 1 PeV (when σ becomes larger than 10 -33 cm 2 ). For neutrino energies of ∼ 100 TeV (10 TeV) the signal is reduced by about 20% (5%).

Taking into account these two effects, the number of muons with energy E µ > E th , crossing an area A during the observation time T can be written as:

N µ = f liv A T ∞ E th dE ν Φ ν (E ν ) Y µ (E ν , E th ) × [1 -āν (E ν )] , (13)
with a similar equation holding for N μ. In Eq. ( 13), āν (E ν ) is the mean coefficient for neutrino absorption through the Earth, including only charged-current interactions and computed for a fixed Earth thickness resulting from averaging over the observation time [19]. Y µ (E ν , E th ) is the muon yield, namely the probability that a neutrino with energy E ν produces a muon with energy E µ > E th that crosses the detector area. This is obtained by integration over the muon energy, E µ , of the neutrino interaction cross section multiplied by the muon range, R(E µ , E th ):

Y µ = ρ H 2 O N A E ν E th dE µ dσ cc dE µ (E ν , E µ ) R(E µ , E th ) . ( 14
)
Here ρ H 2 O is the water density and N A is Avogadro's number. Both the muon range in water and the neutrino interaction probability are taken from [19]. The latter is calculated by using the deep inelastic scattering formula for the charged-current cross-section, σ cc , and the distribution functions of partons as calculated by [29]. In Fig. 3 the solid line shows the resulting N µ + N μ for A = 1 km 2 and T = 1 yr as a function of the energy threshold. For E th = 50 GeV the number of events per year is 4.7, compatible with the findings of [41]. The shaded region represents the muon background produced by atmospheric neutrinos, which can be computed as a function of the energy threshold, E th , by substituting Eq. ( 12) into Eq. ( 13). The result is normalized to an aperture angle of 0.5 • , for the reasons explained above. The uncertainties represented by the shaded region have the same origin as for Fig. 2.

The spectrum of neutrinos from RX J1713.7-3946 dominates the atmospheric background at energies above ∼ 300 GeV. In Table 1 we report the time (in years) of observation needed to obtain a signal with significance level of 3σ, assuming a simple Poisson distribution of events and a unit detection efficiency. One can see that if RX J1713.7-3946 is indeed a hadronic accelerator a signal at the 3σ level could be seen in about 2 years of observation at energies above 500 GeV.

Conclusions

In this paper we described the impact of non linear diffusive shock acceleration for the gamma ray and neutrino production in SNRs. The intricate connection between particle acceleration, magnetic field amplification, dynamical reaction of the particles and magnetic field, and the radiation produced by the accelerated particles (electrons and protons) in principle allow, if taken at face value, to impose strong constraints on the ability of SNRs to accelerate cosmic rays. An instance of how to use this powerful tool was provided in [30], where particle acceleration was described in terms of the non linear theory of Refs. [5,6]. The magnetic field amplification was assumed to be due to resonant streaming instability and both the dynamical reaction of the accelerated particles and of the amplified field were taken into account. The basic conclusion reached in [30] is that the hadronic interpretation of the HESS data automatically leads to a downstream magnetic field which is in agreement with that inferred from X-ray observations if the rims of non-thermal emission are interpreted as the result of synchrotron losses. The non-thermal X-ray spectrum, as measured by Suzaku [38], was also reproduced with unprecedented accuracy. On the other hand, a satisfactory fit to the data within a leptonic scenario could only be achieved by considerably reducing the particle injection efficiency, through fine-tuning of the only free parameter of our calculations, ξ, as defined in §2.1. Moreover, in order for the leptonic model to fit HESS data at all energies one is forced to require the presence of a diffuse background of infrared light exceeding that observed in the interstellar medium by more than a factor 20. These requirements become less stringent if one decides not to include in the fit the highest energy HESS data points, which strongly constrain the maximum energy of the radiating electrons.

Since the discrimination between the two models, hadronic and leptonic, is based on such a tricky region from the observational point of view, we decided to address here the issue of how well the photon energies need to be reconstructed at the telescope in order to tell the difference between the predictions of a leptonic and a hadronic model. The issue is of particular importance since the different predictions are of relevance in the energy region where the particle spectra (and the gamma ray spectrum) are already sharply falling and a small statistical uncertainty in the energy determination may have sizable implications on the shape of the observed spectrum. Our calculations show that a statistical uncertainty in the energy determination larger than about 30% would inhibit the possibility to discriminate between a hadronic and a leptonic interpretation of TeV data. Of course a systematic uncertainty would strengthen the problem. These points need to be taken into account for the design of future telescopes, such as CTA. Needless to say that the observation of TeV emitting SNRs with the Fermi gamma ray telescope will definitely contribute to settle the debate. This should certainly be the case for RX J1713.7-3946 (see also [30]).

In the absence of clear multifrequency evidence however, the smoking gun that SNRs are efficient cosmic ray accelerators can only be provided by the unambiguous detection of neutrinos. Here we used the same non linear theory of particle acceleration to infer the number of neutrino induced muons in a km 3 neutrino telescope. The flux is compared with the appropriate background of atmospheric neutrinos. At the distance of RX J1713.7-3946 and assuming a hadronic interpretation of HESS data, we predict that a 3σ detection should be achieved by km 3 neutrino telescope at the Antares' location (which is in the right hemisphere to detect RX J1713.7-3946) in about 2 years of observation.

e

  -p 2 /p 2 e,max .

Figure 1 :

 1 Figure 1: Hadronic versus leptonic scenarios for the origin of TeV emission from RX J1713.7-3946. In both cases, the flux is computed for the best-fit values of the parameters, as specified in the text. The thick solid line represents the spatially integrated spectral energy distribution of photons from π 0 decay in the hadronic scenario, while the thin solid line refers to ICS in the leptonic scenario. Symbols represent all available HESS data and also plotted are EGRET upper limit and FermiLAT sensitivity for GeV energy photons from this source. Other lines on the plot represent the result of convolution of the thin solid line with gaussians of different widths as explained in the text.

  neutrino telescope, for which the neutrino detection occurs by measuring the Cherenkov light from ν-induced muons produced by charged-current interac-anti-ν µ ν e + anti-ν e

Figure 2 :

 2 Figure 2: Fluxes of photons and neutrinos (plus anti-neutrinos) from RX J1713.7-3946 predicted in the hadronic scenario. Both electronic and muonic neutrinos contributions are shown. The shaded region represents the range of theoretical predictions for the atmospheric muonic neutrino (plus anti-neutrino) flux integrated in a cone of semi-aperture 0.5 • .

Figure 3 :

 3 Figure 3: Neutrino event rate per year from RX J1713.7-3946 (solid line) as a function of muon energy threshold for a neutrino telescope with 1 km 2 effective area. The dashed line shows the event rate expected from atmospheric neutrinos integrated in a cone of semi-aperture 0.5 • .

Table 1 :

 1 Comparison between the expected signal and the atmospheric neutrino background. The last column shows the observation time (in years) required to obtain a signal from RX J1713.7-3946 with a significance level of 3σ.

			µ+ μ	yrs(3σ)
	100	4.4	4.7	2.16
	500	2.8	1.3	1.44
	1000	1.9	0.57	1.41

E th (GeV) N µ+ μ N atm

Acknowledgments

We are grateful to Damiano Caprioli and Gamil Cassam-Chenai for continuous discussion and to Teresa Montaruli for reading the manuscript and providing useful comments. This work was partially supported by MIUR (under grant PRIN-2006) and by ASI through contract ASI-INAF I/088/06/0.


Pierre Samozino
email: pierre.samozino@univ-st-etienne.fr

Jean-Benoît Morin

Frédérique Hintzy

Alain Belli

Jumping ability: A theoretical integrative approach

Keywords:

3) force -velocity relationship 4) performance 5) lower extremity

A c c e p t e d m a n u s c r i p t 2 Summary

A theoretical integrative approach is proposed to understand the overall mechanical characteristics of lower extremities determining jumping ability. This approach considers that external force production during push-off is limited by mechanical constraints imposed by both movement dynamics and force generator properties, i.e. lower extremities characteristics.

While the velocity of the body depends on the amount of external force produced over the push-off, the capabilities of force production decrease with increasing movement velocity, notably for force generators driven by muscular contraction, such as lower extremities of large animals during jumping from a resting position. Considering the circular interaction between these two mechanical constraints, and using simple mathematical and physical principles, the proposed approach leads to a mathematical expression of the maximal jump height an individual can reach as a function of only three integrative mechanical characteristics of his lower extremities: the maximal force they can produce ( 0 F ), the maximal velocity at which they can extend under muscles action ( 0 v ) and the distance of force production determined by their usual extension range ( PO h ). These three integrative variables positively influence maximal jump height. For instance in humans, a 10% variation in 0 F , 0 v or PO h induces a change in jump height of about 10 to 15%, 6 to 11% and 4 to 8%, respectively. The proposed theoretical approach allowed to isolate the basic mechanical entities through which all physiological and morphological specificities influence jumping performance, and may be used to separate the very first macroscopic effects of these three mechanical characteristics on jumping performance variability.

Introduction

Maximal jumps aim to cover the highest vertical and/or horizontal distance from a large acceleration of the body during one quick legs extension. According to Newton's laws of motion, the acceleration of a body during a push-off phase depends on the resultant force applied on it (fundamental principle of dynamics), itself resulting from the force developed by the individual (animal or human) on the environment (law of reciprocal actions).

Consequently, accelerating one's body mass from a resting position to reach the highest jumping performance depends on the production of a large amount of external force during one quick lower extremities extension. Identifying the overall mechanical characteristics of the force generator (i.e. the lower extremities) allowing to maximise this force production over the whole push-off phase could increase the understanding of the mechanical factors limiting jumping performance.

The characteristics of lower extremities related to maximal jumps have been mainly studied through the physiological or morphological traits statistically related to jumping ability (e.g. muscle mass, limbs length, muscle fibre types, joint moment arms), that are the phenotypic expressions of the overall mechanical characteristics of lower extremities involved in performance (see reviews by [START_REF] James | How important are skeletal muscle mechanics in setting limits on jumping performance?[END_REF][START_REF] Marsh | Jumping ability of anuran amphibians[END_REF]. The relationship between jumping performance and overall mechanical characteristics of lower extremities, such as force or power output capabilities, has only been explored through correlations, and in only few studies on humans, which did not reach a consensus [START_REF] Driss | Maximal power and force-velocity relationships during cycling and cranking exercises in volleyball players. Correlation with the vertical jump test[END_REF][START_REF] Yamauchi | Relations between force-velocity characteristics of the knee-hip extension movement and vertical jump performance[END_REF]. Such statistical approaches present the risk of interaction between covariant parameters making uncertain the direct causal effect between dependent and independent variables, though related in some cases by significant correlations. Hence, the large number of biological features that may act on jumping ability (together or independently) make the For these reasons, theoretical approaches have been proposed to better understand the mechanical factors affecting jumping ability. The best examples are forward simulation models that integrate a large amount of morphological, physiological or neuromuscular parameters [START_REF] Alexander | Leg design and jumping technique for humans, other vertebrates and insects[END_REF][START_REF] Bobbert | Effects of muscle strengthening on vertical jump height: a simulation study[END_REF][START_REF] Pandy | Optimal muscular coordination strategies for jumping[END_REF]. These models allowed to answer several questions that could not be solved experimentally, such as optimal muscular coordination strategies or effects of different jumping techniques (e.g. [START_REF] Alexander | Leg design and jumping technique for humans, other vertebrates and insects[END_REF][START_REF] Bobbert | Is the effect of a countermovement on jump height due to active state development?[END_REF][START_REF] Pandy | Optimal muscular coordination strategies for jumping[END_REF]. However, they do not make it possible to isolate the basic mechanical characteristics of lower extremities determining the ability to accelerate the body. Indeed, the numerous inputs used do not represent the overall mechanical characteristics of the force generator, but specific biological features that affect these mechanical characteristics, several of them acting on the same entity.

Other theoretical frameworks have explored these overall mechanical characteristics affecting jumping ability through the investigation of the mechanical requirements of maximal jumps.

Indeed, jumping ballistic and dynamic analyses allow one to identify the mechanical outputs of push-off determining jumping performance at best. For instance, jumping performance has been mathematically shown to be proportional to the mechanical work produced during pushoff (Alexander, 2003a;[START_REF] Emerson | Jumping and Leaping[END_REF] or to the mean power developed before take-off [START_REF] Marsh | Jumping ability of anuran amphibians[END_REF]. These mathematical equations let think that maximal jumping performances are determined by the capability of lower extremities to produce work or power. However, other authors supported the idea that impulsive action performance (such as jumping) do not depend on muscles ability to generate power, but rather on their capability to produce impulse [START_REF] Adamson | Critical Appraisal of Jumping as a Measure of Human Power[END_REF][START_REF] Knudson | Correcting the use of the term "power" in the strength and conditioning literature[END_REF][START_REF] Winter | Jumping: Power or Impulse[END_REF]. Even if this distinction may seem minor, it reveals the difficulty in identifying the mechanical characteristics of the force A c c e p t e d m a n u s c r i p t 6 generator that maximise jumping performance from the sole mechanical outputs related to performance. Whether it is impulse, work or power that determines performance, the question arises as to which mechanical characteristic(s) of lower extremities allow(s) to maximise it.

Further, analyses of the mechanical behaviour of an accelerating body show that all the mechanical outputs characterising push-off dynamics are closely connected [START_REF] Marsh | Jumping ability of anuran amphibians[END_REF][START_REF] Minetti | On the mechanical power of joint extensions as affected by the change in muscle force (or cross-sectional area), ceteris paribus[END_REF][START_REF] Wakai | Optimum take-off angle in the standing long jump[END_REF]. From a theoretical framework, Minetti showed that an increase in the force applied to a mass made the other mechanical outputs of push-off vary: decrease in push-off duration and increase in power and velocity for instance [START_REF] Minetti | On the mechanical power of joint extensions as affected by the change in muscle force (or cross-sectional area), ceteris paribus[END_REF]. Regarding the latter point, and for most of large animals in which skeletal muscles form the basis of the force generator, the increase in movement velocity is associated with a decrease in force production capability [START_REF] Hill | The heat of shortening and the dynamic constants of muscle[END_REF]. Hence, the higher the force production, the faster the movement, and the lower the force production capability. Consequently, take-off velocity, and so jumping performance, depends on the force production of lower extremities over push-off, itself depending on mechanical characteristics of the force generator, themselves depending on push-off velocity. Due to this circular interaction between force generator capabilities and the dynamics of an accelerated body, the mechanical constraints imposed by both force generator and movement dynamics should be taken into account to identify the mechanical characteristics related to maximal jumps.

The aim of this study is therefore to propose an alternative theoretical integrative approach (i) to identify the overall mechanical characteristics of lower extremities determining maximal jumping ability and (ii) to quantify their respective influence by means of the simplest mathematical equation(s) possible. This theoretical framework considers the mechanical constraints of both movement and force generator limiting jumping performance. Only vertical jumping will be considered here since it is an excellent system to investigate jumping abilities.

A c c e p t e d m a n u s c r i p t 7

Theoretical background

The mechanical constraints imposed by both force generator characteristics and movement dynamics on jumping ability are mainly expressed through the interaction between the amount of force produced and the centre of mass (CM) velocity over the whole push-off. The aim of the following sections is to understand how these two mechanical constraints affect the interdependence of force production and movement velocity.

Mechanical constraints imposed by movement dynamics

Jump height (h), which corresponds here to the CM aerial vertical displacement, is dependent on the CM vertical velocity at take-off ( TO v ), according to this basic ballistic equation:

² 2 TO v h g = (1)
This section analyses the push-off phase through the dynamics of the body CM, aiming, with a mathematical approach, to understand how TO v depends on force production. The push-off is considered here in its wholeness, and not studied moment-by-moment. Let consider here a push-off phase during which a body of mass m is accelerated along a vertical direction.

Lower extremities are represented by a linear force generator. The CM velocity (and hence kinetic energy) being null before push-off, the total external mechanical work ( T W ) done by lower extremities during push-off is equal to the variation in the mechanical energy of the body, which is the sum of potential and kinetic energies:

2 1 2 = + T T O P O W mv mgh (2)
with g the gravitational acceleration and PO h the vertical push-off distance corresponding to the extension range of lower extremities. Furthermore, T W , developed by lower extremities

A c c e p t e d m a n u s c r i p t

8 during push-off, is also equal to the product of PO h times the mean vertical external force produced over push-off :

= T PO W F mh (3)
with F the mean vertical external force produced over push-off expressed relative to body mass (in N.kg -1 ). In this form, F has the dimension of an acceleration and would correspond to the component of the CM acceleration induced by the lower extremities propulsive force.

Substituting (2) in this equation gives:

2 2 = + TO PO v F g h (4)
From equation ( 4), TO v may be expressed as:

2 ( ) = - TO PO v h F g (5)
This relationship between TO v , F and PO h represents the mechanical constraints imposed by movement dynamics of an accelerated body. Hence, TO v depends on the mean force produced over push-off, on body mass (since F is expressed relative to body mass) and on the distance through which the force is produced (that is the extension range of lower extremities). The general relationship between TO v and F is presented in Fig. 1 for different values of PO h during a vertical push-off. Each line represents, for different values of PO h , the force-velocity conditions ( F -TO v conditions) allowed by movement dynamics, each F -TO v condition corresponding to one jump height (obtained from equation ( 1)). Jumping performance increases when PO h increases and/or when F increases, F and TO v changing concomitantly in a specific way depending on PO h (equation ( 5)). However, this concomitant increase in F and TO v is limited by the mechanical characteristics of the force generator. Indeed, force -Figure 1 around here -

Mechanical constraints imposed by the force generator

Let consider here only individuals propelling themselves by the direct action of muscles, that is most of large animals [START_REF] Alexander | Leg design and jumping technique for humans, other vertebrates and insects[END_REF]. The mechanical characteristics of the force generator are thus mainly related to the mechanical properties of muscles. The mechanical limits of skeletal muscles capabilities, and notably of the contractile elements, are represented by an inverse force-velocity relationship (F-V relationship), well documented in isolated muscles [START_REF] Hill | The heat of shortening and the dynamic constants of muscle[END_REF] as well as in entire lower extremities musculature during various types of human movement [START_REF] Rahmani | Force/velocity and power/velocity relationships in squat exercise[END_REF][START_REF] Sargeant | Maximum leg force and power output during short-term dynamic exercise[END_REF][START_REF] Vandewalle | Force-velocity relationship and maximal power on a cycle ergometer. Correlation with the height of a vertical jump[END_REF]. This overall inverse relationship, described as linear for pluri-articular movements [START_REF] Bosco | A dynamometer for evaluation of dynamic muscle work[END_REF][START_REF] Rahmani | Force/velocity and power/velocity relationships in squat exercise[END_REF][START_REF] Vandewalle | Force-velocity relationship and maximal power on a cycle ergometer. Correlation with the height of a vertical jump[END_REF][START_REF] Yamauchi | Relations between force-velocity characteristics of the knee-hip extension movement and vertical jump performance[END_REF], accounts for the decrease in the maximal capability for entire lower extremities to generate force with increasing movement velocity. Such linear F-V relationships have widely been showed from mean values of force ( F ) and velocity ( v ) over limbs extension, and have often been defined by two values: the theoretical maximal F that lower extremities can produce over one extension ( 0 F , expressed relative to body mass, in N.kg -1 ) and the theoretical maximal v at which lower extremities can extend during one extension under the influence of muscle action ( 0 v in m.s -1 ). Graphically, 0 F and 0 v correspond to the force-axis and velocity-axis intercepts of the F-V curve, respectively. Thus, during a dynamic maximal effort, such as the push-off phase of a maximal jump, the maximal F that can be produced may be expressed as a function of v :

A c c e p t e d m a n u s c r i p t

10 0 0 0 = - F F F v v (6)
During a maximal jump with a null starting velocity, v and TO v are closely related, as shown in the following demonstration. From Newton's second law of motion, the change in momentum of a body is proportional to the net impulse impressed on this body, and applies along the straight line on which this impulse is impressed. Applying the impulse-momentum relationship to the considered push-off phase with null starting velocity gives:

( ). = - TO PO v F g t (7)
with PO t the push-off phase duration. From equations ( 4) and ( 7), PO t may be expressed as:

2 = PO PO TO h t v (8)
The mean CM vertical velocity v over the whole push-off is the ratio between PO h and PO t , and thus may be obtained from equation ( 8):

2 = TO v v (9)
This equation is only based on the impulse-momentum relationship, and so does not assume that the CM acceleration is constant during push-off, as it was done by [START_REF] Bosco | A simple method for measurement of mechanical power in jumping[END_REF] to

show the same result.

-Figure 2 around here -

Determination of the maximal TO v that can be reached

Considering both the mechanical constraints imposed by the movement itself ( TO v increasing with F , equation ( 5)) and those imposed by the force generator (force production capability decreasing with increasing velocity, equation ( 6)) allows to determine the maximal F that can be produced during a vertical push-off, maximising TO v and h. ) that can be reached is achieved in the push-off condition allowed by both force generator maximal capabilities and movement dynamics. Such on optimal condition corresponds graphically to the couple of force and velocity values at the two lines intersect.

Consequently, the force and velocity values characterising this optimal condition have to be a valid set of solutions for the system formed by equations ( 5), ( 6) and ( 9), the unknown variables being F , v and TO v . Solving this system of equations, which includes a non-linear equation (equation ( 5)), gives two quadratic equations each including F or TO v . The quadratic equation including TO v has only one real positive solution for TO v ( max TO v ) that may be expressed as:

0 0 max 0 0 0 ² 2 ( ) 4 ² 2 ⎛ ⎞ = + -- ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ TO PO PO F F v h F g v h v (10)
From equation 1, the maximal jump height that can be reached ( max h ) may be expressed as:

0 0 max 0 0 0 ² ² ² 2 ( ) 2 4 ² 2 ⎛ ⎞ = + -- ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ PO PO F F h h F g g v h v ( 11
)
From these computations, max h that could be conceivable respecting the mechanical constraints of both movement and force generator depends only on 0 F , 0 v and PO h , which are

Results

The influences of 0 F , 0 v and PO h on max h were analysed from equation ( 11). First, a sensitivity analysis was performed plotting the relative variations of max h against the relative variations of 0 F , 0 v and PO h , each parameter being studied separately (Fig. 3). The reference values of the three parameters corresponded to human-like values ( 0

F =30 N.kg -1 , 0 v =3 m.s -1
and PO h =0.4 m; e.g. [START_REF] Bosco | A dynamometer for evaluation of dynamic muscle work[END_REF][START_REF] Rahmani | Force/velocity and power/velocity relationships in squat exercise[END_REF], but the range of variations considered allowed a wide variety of large animals to be concerned. Maximal jump height increases when 0 F , 0 v or PO h increase, and it is more sensitive to 0 F than to 0 v or PO h .

Indeed, changes in max h with 0 F variations are higher than those induced by 0 v variations, themselves higher than those induced by PO h variations: a 10% change in 0 F leads to a ~10 to 15% change in max h , while the same 10% change in 0 v and PO h leads to changes in max h ranging from 6 to 11 % and from 4.5 to 7.5%, respectively. Note that these relative influences were calculated from reference values corresponding to human characteristics, and may be different for other reference values.

-Figure 3 -Figure 4 around here -

Considering the few values of 0 F (between 25 and 50 N.kg -1 ) and 0 v (between 1.5 and 8 m.s - 1 ) reported in previous human studies [START_REF] Rahmani | Differences in morphology and force/velocity relationship between Senegalese and Italian sprinters[END_REF][START_REF] Rahmani | Force/velocity and power/velocity relationships in squat exercise[END_REF][START_REF] Yamauchi | Relations between force-velocity characteristics of the knee-hip extension movement and vertical jump performance[END_REF] and estimating PO h between 0.35 and 0.45 m, max h values obtained range from ~0.2 to ~0.6 m, which is in line with human vertical jumping performance (Fig. 4, the maximal values of each parameter being unlikely obtained by a same given subject). Further, the validity of equation ( 11) has been confirmed by experimental data comparing actual and predicted values of max h in human vertical jumps (unpublished personal data).

Discussion

This study proposes a novel mathematical approach to identify the mechanical characteristics of lower extremities involved in maximal jumping ability and to quantify their respective influence on this ability. Considering the mechanical constraints of both movement and force generator that limit the force production during push-off, this integrative approach leads to a mathematical expression of the maximal jump height an individual can reach ( max h ) as a function of only three mechanical entities that represent overall mechanical characteristics of lower extremities: the theoretical maximal dynamic force that can be generated over one lower extremities extension ( 0 F ), the theoretical maximal velocity at which the lower extremities can extend under the influence of muscles contraction in unloaded conditions ( 0 v ), and the extension range of the lower extremities determining the distance over which force can be generated ( PO h ).

A c c e p t e d m a n u s c r i p t 14

Limits of the theoretical integrative approach

The theoretical developments proposed to express max h as a function of only three mechanical characteristics of lower extremities required many simplifying assumptions that could limit their application. First, since the mechanical properties of the force generator have been characterised by an inverse F-V relationship, the present form of equation ( 11) only applies to animals propelling themselves by the direct action of muscles. Further, the linearity of the F-V relationship of the entire lower extremities musculature, as well as the extrapolation of 0 F and 0 v , is well admitted for human lower limbs extension [START_REF] Rahmani | Force/velocity and power/velocity relationships in squat exercise[END_REF][START_REF] Yamauchi | Relations between force-velocity characteristics of the knee-hip extension movement and vertical jump performance[END_REF]), but has not been thoroughly investigated in animals, mainly for obvious practical reasons. It is noteworthy that 0 F and 0 v are two purely theoretical values and have to be considered as target values towards which maximal capabilities of the force generator tend.

Hence, 0 F and 0 v have to be understood as the "force" and "velocity" maximal capabilities of the entire lower extremities. Other reasonable assumptions have been made, such as those related to the application of Newton's laws to a whole body considered as a system or the neglected air resistance which only affects very small animals jump height (body mass below 0.5 g, Scholz et al., 2006a). Moreover, it was supposed that the CM vertical displacement during push-off corresponded to the extension range of the lower extremities (i.e. PO h ), whereas the actual relative position of the CM within the body shifts slightly downwards due to this lower extremities extension. This overestimation has nevertheless minor effects on mechanical outputs estimation during a jumping push-off [START_REF] Samozino | A simple method for measuring force, velocity and power output during squat jump[END_REF]. The main limit of the proposed approach remains in the relationship, existing in some cases, between Consequently, the effect of intra-individual variations in PO h on max h has to be cautiously considered since PO h variations may involve changes in 0 F and 0 v . Therefore, comparisons of jumping ability for a given individual at two different instants (e.g. before and after training) require to keep PO h constant [START_REF] Minetti | On the mechanical power of joint extensions as affected by the change in muscle force (or cross-sectional area), ceteris paribus[END_REF][START_REF] Zamparo | Interplay among the changes of muscle strength, cross-sectional area and maximal explosive power: theory and facts[END_REF]. However, for inter-individual or inter-species comparisons, the considered PO h has to correspond to the optimal or usual one for each individual, which allows to investigate differences in maximal performance between individuals. In the following part of this discussion, when the effect of PO h on max h is considered, one should consider several individuals with different usual or optimal PO h , and not the effect of intra-individual changes in PO h .

Jumping ability is explained by only three integrative mechanical variables

The originality of the proposed approach is to explain jumping performance from the fewest variables possible, which agrees for instance with Alexander's conception of mathematical models in biological systems: "the simpler the model, the clearer it is which of its characteristics are essential to the observed effect" (Alexander, 2003b). The whole point of explaining jumping performance by the fewest lower extremities mechanical characteristics possible is to isolate the basic mechanical entities through which all physiological and morphological specificities influence jumping performance. The three integrative variables ( 0 F , 0 v and PO h ) identified in equation ( 11) do not correspond to phenotypic traits, but

A c c e p t e d m a n u s c r i p t

16 represent the overall mechanical characteristics of the entire lower extremities. This implies these variables to be integrative, and the explored resulting effects to be macroscopic. These three mechanical characteristics encompass all the morphological and physiological parameters previously put forward to explain the variability in jumping performance. For instance, 0 F and 0 v do not only correspond to intrinsic muscles properties, but are the resultant of all the biological features affecting the maximal force that can be developed during lower extremities extension and the maximal extension velocity, respectively.

The positive influence of 0 F on max h is in line with studies relating jumping ability to muscle strength [START_REF] Alexander | Leg design and jumping technique for humans, other vertebrates and insects[END_REF][START_REF] Cheng | The relationship between joint strength and standing vertical jump performance[END_REF][START_REF] Maffiuletti | Effect of combined electrostimulation and plyometric training on vertical jump height[END_REF]Scholz et al., 2006b;[START_REF] Ugrinowitsch | Influence of training background on jumping height[END_REF][START_REF] Yamauchi | Relations between force-velocity characteristics of the knee-hip extension movement and vertical jump performance[END_REF] or animals hind limb muscle mass [START_REF] Choi | Variations in take-off velocity of anuran amphibians: Relation to morphology, muscle contractile function and enzyme activity[END_REF][START_REF] James | Explosive jumping: extreme morphological and physiological specializations of Australian rocket frogs (Litoria nasuta)[END_REF][START_REF] James | How important are skeletal muscle mechanics in setting limits on jumping performance?[END_REF]. Indeed, maximal muscle force, which is proportional to muscles cross-sectional areas [START_REF] Alexander | The maximum forces exerted by animals[END_REF], logically increases with the amount of muscle mass involved in push-off. Other factors previously related to jumping performance directly affect 0 F : a high rate of force development [START_REF] Vanezis | A biomechanical analysis of good and poor performers of the vertical jump[END_REF], an optimised motor unit recruitment [START_REF] Maffiuletti | Effect of combined electrostimulation and plyometric training on vertical jump height[END_REF] or positions of muscles origin and insertion increasing torque and force production [START_REF] Emerson | Jumping and Leaping[END_REF]. Besides, it is worth noting that 0 F represents a force normalised to body mass (expressed in equation ( 11) in N.kg -1 ). Since larger animals are known to produce lower maximal forces relative to their body mass (and thus lower 0 F ), max h is logically influenced by animals body size, as previously reported [START_REF] Alexander | The maximum forces exerted by animals[END_REF][START_REF] James | How important are skeletal muscle mechanics in setting limits on jumping performance?[END_REF]Scholz et al., 2006a).

Similarly to 0 F , the positive influence of 0 v on max h is in accordance with studies showing that frog muscles used to power jumping are mainly composed of fast-twitch muscle fibres [START_REF] Lutz | Quantitative analysis of muscle fibre type and myosin heavy chain distribution in the frog hindlimb: implications for locomotory design[END_REF][START_REF] Marsh | Jumping ability of anuran amphibians[END_REF], and with studies relating jumping performance to muscle strength only at high movement velocity [START_REF] Eckert | Angular velocity and range of motion in the vertical and standing broad jumps[END_REF]. The effect of 0 v on jumping favouring in many cases the velocity of movement [START_REF] Emerson | Jumping and Leaping[END_REF]. Neuromuscular coordination strategies, widely presented as determinant for vertical jumping performance [START_REF] Bobbert | Effects of muscle strengthening on vertical jump height: a simulation study[END_REF][START_REF] Pandy | Optimal muscular coordination strategies for jumping[END_REF], are also integrated in the mechanical characteristics 0 F and 0 v . The same is true for the positive effect of PO h on max h . Indeed, PO h includes numerous morphological characteristics that increase the range of motion and have previously been related to jumping performance, such as hind limb length [START_REF] Choi | Variations in take-off velocity of anuran amphibians: Relation to morphology, muscle contractile function and enzyme activity[END_REF][START_REF] Emerson | Jumping and Leaping[END_REF][START_REF] Harris | The relationship between maximum jumping performance and hind limb morphology/physiology in domestic cats (Felis silvestris catus)[END_REF][START_REF] James | Explosive jumping: extreme morphological and physiological specializations of Australian rocket frogs (Litoria nasuta)[END_REF], the number of joints and their angular range of motion [START_REF] Alexander | Leg design and jumping technique for humans, other vertebrates and insects[END_REF], or the height of the starting position [START_REF] Alexander | Leg design and jumping technique for humans, other vertebrates and insects[END_REF][START_REF] Domire | The influence of squat depth on maximal vertical jump performance[END_REF][START_REF] Selbie | A simulation study of vertical jumping from different starting postures[END_REF].

Interestingly, James Gray already put forward that it was not the leg length itself that is advantageous in jumping, but the ability to extend the limb to a greater length [START_REF] Gray | Jumping and Creaping, How animals move[END_REF].

Respective influences of 0 F , 0 v and PO h on max h

The three mechanical characteristics 0 F , 0 v and PO h , and all the above mentioned biological characteristics that positively influence them, have a positive effect on jumping ability. This is obvious when referring to Figs. 1 and 2: increasing max h may be achieved by shifting the F-V relationship of lower extremities to the right (i.e. increasing both 0 F and 0 v ) and/or by shifting the curve of the F -TO v conditions allowed by movement dynamics upwards (i.e.

increasing PO h ). Simulations performed with equation ( 11) show that max h is more dependent on 0 F or 0 v than on PO h . As a result, maximal jump seems to be more limited by the mechanical characteristics of force generator (i.e. F-V relationship) than by movement dynamics depending on PO h . When trying to differentiate the respective weight of 0 F and 0 v , simulations performed from human-like reference values (sensitivity analysis in Fig. 3) seem seems to be enhanced with the increase in the other variables, as shown by the shape of contours in Fig. 4. Consequently, jump height of smaller animals, known to produce larger relative maximal force [START_REF] Alexander | The maximum forces exerted by animals[END_REF], more sensitive to 0 v changes than larger animals jump height is, which agrees with Alexander's model simulations [START_REF] Alexander | Leg design and jumping technique for humans, other vertebrates and insects[END_REF]. In the same way, in hypogravity conditions, in which the relative maximal force of an individual is higher than on the Earth, the influence of 0 v on jump height would increase to the detriment of the influence of 0 F , which is in line with Minetti's conclusions [START_REF] Minetti | On the mechanical power of joint extensions as affected by the change in muscle force (or cross-sectional area), ceteris paribus[END_REF].

The mathematical expression of jumping performance (equation ( 11)) provides a theoretical framework to separate the very first macroscopic effects of 0 F , 0 v and PO h on jumping performance variability. Adopting the same reasoning as [START_REF] Minetti | On the mechanical power of joint extensions as affected by the change in muscle force (or cross-sectional area), ceteris paribus[END_REF], equation ( 11 and/or 0 v ), and notably, as argued by the authors, by a higher specific force (maximal force per unit muscle mass).

Other strategies to improve jumping abilities

In addition to biological adaptations, individuals also used specific techniques that allow higher maximal jump height by overcoming the muscular constraints. The most famous one in humans is the countermovement performed just before push-off, leading to 2 to 4 cm higher jumps [START_REF] Bobbert | Is the effect of a countermovement on jump height due to active state development?[END_REF]. This greater performance has been firstly explained by effects of elastic energy and myoelectrical potentiation [START_REF] Bosco | Combined effect of elastic energy and myoelectrical potentiation during stretch-shortening cycle exercise[END_REF], and more recently by the only fact that muscle active state develops during the preparatory downward movement, which involves a higher force at the beginning of push-off, and so improves the maximal force capability [START_REF] Bobbert | Is the effect of a countermovement on jump height due to active state development?[END_REF]. In Fig. 2, the improvement in jump height thanks to a countermovement may be graphically represented by a shift of the lower extremities F-V relationship to the right, that is by an increase in 0 F and/or 0 v [START_REF] Bosco | Potentiation of the mechanical behavior of the human skeletal muscle through prestretching[END_REF][START_REF] Bosco | Combined effect of elastic energy and myoelectrical potentiation during stretch-shortening cycle exercise[END_REF]. Another strategy has been observed in small animals to face with morphological or muscular constraints, such as short hind limbs limiting PO h [START_REF] Burrows | The effect of leg length on jumping performance of short-and long-legged leafhopper insects[END_REF][START_REF] James | How important are skeletal muscle mechanics in setting limits on jumping performance?[END_REF] or poor muscle maximum shortening velocity reducing 0 v [START_REF] Alexander | Leg design and jumping technique for humans, other vertebrates and insects[END_REF][START_REF] James | How important are skeletal muscle mechanics in setting limits on jumping performance?[END_REF]. Such a jump technique, particularly described in insects [START_REF] Bennet-Clark | The energetics of the jump of the locust Schistocerca gregaria[END_REF] and commonly called "catapult mechanism" [START_REF] Alexander | Leg design and jumping technique for humans, other vertebrates and insects[END_REF][START_REF] James | How important are skeletal muscle mechanics in setting limits on jumping performance?[END_REF], allows animals to overcome skeletal muscles limits by storing elastic energy and releasing it suddenly, that is more quickly than what muscles alone could do [START_REF] Alexander | Leg design and jumping technique for humans, other vertebrates and insects[END_REF][START_REF] Bennet-Clark | The energetics of the jump of the locust Schistocerca gregaria[END_REF][START_REF] Burrows | The effect of leg length on jumping performance of short-and long-legged leafhopper insects[END_REF]. In many small insects, in [START_REF] Bennet-Clark | The energetics of the jump of the locust Schistocerca gregaria[END_REF], jumping performance is not limited by muscle contractile properties but rather by the mechanical characteristics of elastic energy storage and recoil. Considered as a linear spring, the series elastic elements can therefore be assumed to shorten at unlimited velocities [START_REF] Alexander | Leg design and jumping technique for humans, other vertebrates and insects[END_REF], the corresponding 0 v value tending then to infinite. Simple computations (from data reported by [START_REF] Bennet-Clark | The energetics of the jump of the locust Schistocerca gregaria[END_REF] and [START_REF] Alexander | Leg design and jumping technique for humans, other vertebrates and insects[END_REF]) may show that the maximal jump height a locust can reach using the catapult mechanism may be estimated to ~50 cm while muscles contraction alone would lead to jumps not higher than 5 cm. With the "catapult mechanism", the muscle F-V relationship is not a mechanical limit to jumping performance, which is in line with the low influence of the muscle maximal shortening velocity on the "catapult" jump height [START_REF] Alexander | Leg design and jumping technique for humans, other vertebrates and insects[END_REF]. Other animals, larger than insects, have been shown to also use "catapult-like" mechanisms to jump, namely frogs (e.g. [START_REF] James | How important are skeletal muscle mechanics in setting limits on jumping performance?[END_REF] or bushbabies (e.g. [START_REF] Aerts | Vertical jumping in galago senegalensis: the quest for an obligate mechanical power amplifier[END_REF]. Even if the mechanical work done during push-off is provided by both elastic energy release and muscles contraction, this jump technique, also qualified as a "mechanical power amplifier" [START_REF] Aerts | Vertical jumping in galago senegalensis: the quest for an obligate mechanical power amplifier[END_REF], largely improves jumping performance by increasing the "velocity" capabilities of the force generator.

Conclusion

This study proposed a new integrative approach leading to a mathematical expression of the maximal jump height an individual can reach from only three mechanical characteristics of its lower extremities: their maximal capability of force production, their maximal extension velocity and their usual extension range. This theoretical approach increases then the understanding of the mechanical factors limiting jumping performance. The main originality of this mathematical expression is to isolate the basic mechanical entities through which all physiological and morphological specificities influence jumping performance, and to quantify Changes in the take-off velocity and jump height as a function of the mean force produced over vertical jump push-off for different push-off distances ( PO h ). Each line represents the Force-Velocity conditions allowed by movement dynamics during a given push-off. Jump height and take-off velocity are related by equation ( 1). Since force is expressed relative to body mass, the x-axis also corresponds to the component of the CM acceleration induced by the propulsive force.

Figure 2

Theoretical representation of the mechanical constraints imposed by movement dynamics (continuous line) and lower extremities capabilities (dashed line) during a typical vertical jump. The continuous line represents the take-off velocity according to the mean force produced over push-off. The Force -Velocity conditions (mean force and mean velocity over push-off) allowed by lower extremities mechanical capabilities are represented for sub maximal efforts (grey area) and for maximal explosive effort (dashed line). Here, PO h , 0 F and 0 v values are 0.4 m, 30 N.kg -1 and 3 m.s -1 , respectively.

F

  to the vertical distance covered by the CM during the aerial phase (in m) T W total external mechanical work produced by lower extremities during push-off (J) PO h vertical push-off distance determined by lower extremities extension range (in m) TO v CM vertical velocity at take-off (in m.s -1 ) F mean vertical external force developed over push-off (relative to body mass, in N.kg -1 ) v mean vertical CM velocity over push-off (in m.s -1 ) 0 theoretical maximal value of F that lower extremities can produce during one extension at a theoretical null v (relative to body mass, in N.kg -1 ) 0 v theoretical maximal value of v at which lower extremities can extend during one extension under the influence of muscles action in a theoretical unloaded condition (in m.s -1 ) PO t push-off time (in s) max TO v maximal CM vertical take-off velocity a given animal can reach (m.s -1 ) max h maximal jump height a given animal can reach (m)

  lower extremities characteristics involvement in such abilities difficult on the sole basis of experimental data.

  well known to depend on velocity for force generators composed by skeletal muscles.

  Fig.2presents these two constraints acting on force production during a vertical push-off for a given human-like PO h (0.4 m). The grey area under the F-V relationship (dashed line)represents the whole Fv conditions that lower extremities musculature can develop, the F-V relationship being the F -v conditions corresponding to maximal efforts. The solid line represents, as in Fig.1, the F -TO v conditions allowed by the movement dynamics, higher jumping performance being reached when F increases. Consequently, the maximal value of TO v ( max TO v

  h values were chosen to correspond to most of large animals, jumping by means of direct muscles contraction[START_REF] Alexander | Leg design and jumping technique for humans, other vertebrates and insects[END_REF]. The positive influence of the three parameters on max h is confirmed for a wide range of values. Moreover, the shape of contours influence of each parameter is enhanced by the increase in the two other parameters.

v

  , as mean values representing maximal force and velocity over the whole push-off, are specific to the lower extremities range of extension ( PO h ). For a given individual, a change in for instance by a change in starting position) may lead to variations in 0 F and 0 v due to the effects of both muscles force-length relationships and changes in joint moment arms during extension. Hence, 0 F and 0 v correspond to the maximal force and velocity capabilities of lower extremities for a given extension range, i.e. for a given PO h .

  been suggested through the positions of muscles insertion around joints,

  maximal jumping ability is more limited by 0 F than by 0 v . Thus, force capabilities seem to represent the mechanical entity playing the most important role in jumping performance, at least in humans. The respective weight of either 0 F or 0 v on max h

  ) could be used to separate the effects of intra-individual changes in force or velocity capabilities when a change in jumping performance is observed. Conversely, one could use equation(11) to predict jump height changes induced by expected modifications (positive or negative) in force generator capabilities, which may guide rehabilitation and/or training programs. On the other hand, such an integrative approach could also form a first stage to guide further explorations of underlying biomechanical and physiological mechanisms at lower levels of organization. For instance, a simple application of equation (11) makes it possible to explore the differences in jumping ability between humans and bonobos from data presented byScholz et al (2006b). If 0 F and 0 v are similar, the observed difference of maximal jump height between the man (0.34 m) and the bonobo (0.52 m estimated without arm movement effect) may be explained for ~40% (i.e. ~0.07 m) by the difference in PO h (~0.42 m for the .65 m for the bonobo from their optimal starting position). The remaining ~60% of difference (i.e. ~0.11 m) may be then explained by better force generator characteristics ( 0 F

  amount of energy required to jump can be stored in elastic structures

  FIGURES CAPTIONS

Figure 3

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  Figure 4

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