Analytical Solution for the Deformation of a Cylinder under Tidal Gravitational Forces

Quite a few future high precision space missions for testing Special and General Relativity will use optical resonators which are used for laser frequency stabilization. These devices are used for carrying out tests of the isotropy of light (Michelson-Morley experiment) and of the universality of the gravitational redshift. As the resonator frequency not only depends on the speed of light but also on the resonator length, the quality of these measurements is very sensitive to elastic deformations of the optical resonator itself. As a consequence, a detailed knowledge about the deformations of the cavity is necessary. Therefore in this article we investigate the modeling of optical resonators in a space environment. Usually for simulation issues the Finite Element Method (FEM) is applied in order to investigate the influence of disturbances on the resonator measurements. However, for a careful control of the numerical quality of FEM simulations a comparison with an analytical solution of a simplified resonator model is beneficial. In this article we present an analytical solution for the problem of an elastic, isotropic, homogeneous free-flying cylinder in space under the influence of a tidal gravitational force. The solution is gained by solving the linear equations of elasticity for special boundary conditions. The applicability of using FEM codes for these simulations shall be verified through the comparison of the analytical solution with the results gained within the FEM code. Deformation of a Cylinder under Tidal Gravitational Forces 2 1. Motivation Special (SR) and General Relativity (GR) are two of the most important theories and theoretical frames of modern physics. They are the basis for the understanding of space and time and thus for the underlying physical structure of any other theory. The interest in testing the fundamentals of SR and GR has grown enormously over the last years as all presently discussed approaches to quantum gravity predict tiny violations of SR and GR. The technological improvements of the last decades have provided scientists with high precision measurement equipment such as optical resonators. In optical resonators laser locking is used to define stable optical frequencies. The resonance frequency of the locked lasers is given by ν = mc/L where c is the speed of light, L the resonator length L, and m the mode number. Optical resonators have been used recently to test one of the pillars of Special Relativity, namely the isotropy of the speed of light [7, 8] as well as of the universality of the gravitational redshift [5]. In doing so, two laser beams are locked to two orthogonally oriented resonators. An anisotropic speed of light would lead to a beat of the frequencies during a rotation of this setup. Due to the importance of this type of experiments one looks for ways to improve that. One option for this is to carry out these experiments in space, as planned in the OPTIS mission [19] or with SUMO on the ISS [13]. Although many of the disturbances acting on a resonator can be minimized by means of an appropriate satellite control system, some intrinsic disturbances cannot be eliminated as a matter of principle and distort the resonator shape leading to a systematic frequency shift. In particular the tidal gravitational force ‡ which acts through every extended body cannot be eliminated by choosing an appropriate frame and, thus, will induce distortions on the resonator. We give a rough estimate of the expected effect of the tidal gravitational force on a freely moving cube of length L. If the position of the cube is at a distance R from the center of the Earth, then the difference of the Earth’s acceleration on the top and bottom of the cube is ∆a = (∂U/∂r)L, where U is the Earth’s Newtonian potential U = GM⊕/R. For an orbit with R = 10000 km and a typical resonator length of L = 5 cm we have ∆a ≈ GM⊕/R L ≈ 2 · 10 m/s. In a rough estimate we assume this ∆a to act on the top surface of the cube. Now Hook’s simple law of elasticity F A = E ∆L L (1) gives the change of the length ∆L of the cube due to a force F acting on the area A. In our case taking F = m∆a = ρL∆a we get ∆L L = ρL E ∆a ≈ 10 (2) assuming an elasticity modulus of E = 90 GPa and a density of 2350 kg/m 3 which is typical for Zerodur. In the OPTIS mission, for example, the science goal for the measurement of the isotropy of the speed of light is better than ∆c/c = 10 [19]. This can only be achieved if the resonator has a length stability of ∆L/L = 10 [19]. As one can see from our estimates, the tidal gravitational force will lead to systematic deformations ‡ In space and engineering sciences the tidal gravitational force is often referred to as ’gravity gradient’. Deformation of a Cylinder under Tidal Gravitational Forces 3 which are one order of magnitude larger than the expected accuracy. Therefore the effect has to be investigated carefully by including the tidal gravitational force into the equations of elasticity, calculating the resulting resonator shape, and then subtract the effect. Although the linear theory of elasticity has a long history, explicit solutions for special problems are purely spread. In textbooks only examples for simple bodies in homogeneous gravitational fields or for thermal expansions can be found (e.g. [14], [15], [18]). However, most of the solutions employ an ansatz which already includes knowledge about the expected solution. To the understanding of the authors, no publications are available dealing with a body under the influence of a tidal gravitational force so far. The reason for this is probably, that this situation applies only to bodies freely flying in space – a situation which was outside the scope of application in elasticity theory so far. In the present paper we first derive an analytical solution in terms of a series expansion. This result is then confirmed using numerical methods. These calculations are usually done with help of Finite Element Method (FEM) codes. For most engineering purposes FEM codes are fine. However FEM solutions are only numerical approximations whose accuracy depends highly on the number and shape of the elements that have been chosen to mesh the model. In order to confirm the analytical model and to test the numerical calculation, we compare the analytical with the numerical solution. For this comparison we choose a cylinder as most simple geometry of a body adapted to the symmetry of the problem. Having thus checked the principal applicability of the FEMmethods to these kinds of physical situations, this method safely can be used for calculating the deformations of arbitrarily shaped bodies or for the design of devices insensitive to unwanted influences, or for the elimination of the systematics of the measurements in order to ensure the success of highly sensitive experiments. 2. Basic Equations 2.1. Generalities The problem of an optical resonator flying on a geodetic Earth orbit can be simplified by treating the problem in a body fixed coordinate system. We also consider, for simplicity, the body to be a homogeneous and isotropic cylinder. The only force present is a volume force due to the tidal gravitational force which will be modeled as gradient of a spherically symmetric Earth acceleration field. In order to calculate the elastic deformations of the cylinder the equations of elasticity have to be solved including the influence of the tidal gravitational force. The boundary conditions for the solution are given through the condition of weightlessness in space. As a short introduction, some basic equations of the linear theory of elasticity are given [15, 14, 18, 11]. All equations refer to homogeneous isotropic bodies. Within this paper we do not use the notation within the formalism of the Riemannian geometry (e.g. [20]) but the notation used in [15]. In elasticity the general relation between the stress tensor σij and the strain tensor εij is given by Hooke’s law σij = Cijklεij i, j, k, l = 1, 2, 3 (3) Deformation of a Cylinder under Tidal Gravitational Forces 4 where Cijkl is the elasticity tensor related to the material under consideration. For homogeneous isotropic materials the elasticity tensor can be written as Cijkl = λδijδkl + μ(δikδjl + δilδjk) , (4) where λ and μ are the Lamé constants and δmn is the Kronecker symbol. Thus Hook’s law for homogeneous isotropic materials is σij = λδijεkk + 2μεij . (5) The strain tensor ε has to fulfill the so-called compatibility condition ∂εil ∂rj∂rk + ∂εjk ∂ri∂rl − ∂εjl ∂ri∂rk − ∂εik ∂rj∂rl = 0 , (6) where rn are the components of the position vector. The relations between strain and the displacement ξi are εij = 1 2 ( ∂ξi ∂rj + ∂ξj ∂ri ) . (7) The equilibrium equation of elasticity describes the equilibrium state of a homogeneous isotropic body when a volume force ~ K is acting ∂ ∂rj σij +Ki = 0 . (8) Applying the relations between stress and displacements the equilibrium equation takes the form [15] (λ + μ) ∂ ∂rk∂rj ξk + μ ∂ ∂ri∂ri ξj +Kj = 0 , (9) where ~ ξ is the displacement vector. This equation can also be written as ∆~ ξ + 1 1− 2ν ∇(∇ · ~ ξ) + 1 μ ~ K = 0 (10) where ν is the Poisson number which lies between 0 and 0.5 for homogeneous isotropic bodies. For vanishing volume forces ~ K = 0 Eq. (9) becomes the homogeneous equilibrium equation (λ + μ) ∂ ∂rk∂rj ξk + μ ∂ ∂ri∂ri ξj = 0 . (11) The boundary conditions for the solution of the equilibrium equation are either given by the forces pi acting on the body surfaces σijnj = pi (12) or by initial displacements ξi0 of the surfaces ξi(0) = ξi0 (13) where nj are the normal vectors on the surfaces. The general solution of Eq. (10) is a superposition of a homogeneous and a particular solution ~ ξ = ~ ξ + ~ ξ . (14) Deformation of a Cylinder under Tidal Gravitational Forces 5 2.2. The symmetries of our problem Since we have an axial symmetric problem, we use cylindrical coordinates r, φ, z is useful. All displacements and derivatives with respect to φ vanish and the equilibrium equation of elasticity takes the form (see e.g. [15]) 0 = ∆ξr − ξr r2 + 1 1− 2ν ∂ ∂r ( ∂ξr ∂r + ξr r + ∂ξz ∂z ) 0 = ∆ξz + 1 1− 2ν ∂ ∂z ( ∂ξr ∂r + ξr r + ∂ξz ∂z ) . (15) The Laplace operator acting on a scalar takes the form ∆ = ∂ ∂r2 + 1 r ∂ ∂r + ∂ ∂z2 . (16) Note that the Laplace operators acting on a vector field ~ ξ takes the form (see [12]) ∆ξr = ∂ξr ∂r2 + 1 r ∂ξr ∂r + ∂ξr ∂z2 − ξr r2 ∆ξz = ∂ξz ∂r2 + 1 r ∂ξz ∂r + ∂ξz ∂z2 . (17) The relations between stresses, strains and displacements are σrr = λ(εrr + εφφ + εzz) + 2μεrr = λ ( ∂ξr ∂r + ξr r + ∂ξz ∂z ) + 2μ ∂ξr ∂r σφφ = λ(εrr + εφφ + εzz) + 2μεφφ = λ ( ∂ξr ∂r + ξr r + ∂ξz ∂z ) + 2μ ξr r σzz = λ(εrr + εφφ + εzz) + 2μεzz = λ ( ∂ξr ∂r + ξr r + ∂ξz ∂z ) + 2μ ∂ξz ∂z σrφ = 0 σφz = 0 σrz = 2μεrz = μ ( ∂ξr ∂z + ∂ξz ∂r ) . (18) Beside the axial symmetry we also have the following symmetries for reflection at the z = 0 plane: ξz(r,−z) = −ξz(r, z) and ξr(r,−z) = ξr(r, z). 3. The Problem In order to solve the problem of a free-flying isotropic homogeneous cylinder in space the equilibrium equation of elasticity (10) has to be solved. The cylinder has radius R and height 2L. The body coordinates are (r, φ, z) with the origin being at the center–of–mass of the cylinder. The z–axis coincides with the symmetry axis of the cylinder, see Fig. 1. The only force present is the volume force ~ K which is due to the Earth’s gravitational potential U , ∆~ ξ + 1 1− 2ν ∇(∇ · ~ ξ)− 1 μ ρ∇U = 0 . (19) Deformation of a Cylinder under Tidal Gravitational Forces 6 Figure 1. Simplified model of an optical resonator on a geodetic Earth orbit For a spherical Earth potential, U(r) = GM⊕/r, where GM⊕ is the gravitational constant times the mass of Earth, the potential acting at an arbitrary point P inside the cylinder can be calculated via Taylor expansion U(~rM + ~r) = U(~rM ) + ∂U(~rM ) ∂ri ri + 1 2 ∂U(~rM ) ∂ri∂rj rirj (20) = U(~rM ) +∇U(~rM )~r + GM⊕ 2r3 M (r − 2z) (21) where ~rM is the vector from the center–of–mass of the Earth to the center–of–mass of the cylinder and ~r is the vector from the cylinder center–of–mass to point P . This Taylor expansion around the center–of–mass of the cylinder to second order gives the axis-symmetric potential in cylindrical coordinates. Note that the linear term of the Taylor expansion vanishes as this equation is valid in the freely falling reference frame of the cylinder. Since we consider a freely flying cylinder in an orbit around the Earth, no external forces are present and, thus, the forces ~ p at the cylinder surfaces are zero which gives us the boundary conditions σijnj = pi = 0 . (22) The normal vector ~n = (nr nφ nz) T (T means the transposed vector) reduces in the axis-symmetric case to ~n = (nr 0nz) T , as the φ component is zero. Thus the boundary conditions (22) at the top surface of the cylinder, i.e. for z = L, ~n = (0 0 1) , are pr(r, z = L) = 0 = σrz(r, z = L) pz(r, z = L) = 0 = σzz(r, z = L) . (23) At the bottom surface of the cylinder, i.e. for z = −L, ~n = (0 0−1) , we have pr(r, z = −L) = 0 = −σrz(r, z = −L) pz(r, z = −L) = 0 = −σzz(r, z = −L) . (24) Deformation of a Cylinder under Tidal Gravitational Forces 7 Note that these boundary conditions are valid for arbitrary r ∈ [0, R). For r = R they are not valid as the normal vector is not uniquely defined at the cylinder edges (r, z) = (R,±L). For the superficies cylinder surface, i.e. r = R, ~n = (1 0 0) , the boundary conditions are pr(r = R, z) = 0 = σrr(r = R, z) pz(r = R, z) = 0 = σzr(r = R, z) (25) for all z ∈ [0,±L). 4. The Solution 4.1. Particular Solution In order to find a particular solution of the problem one can assume that the solution of the equilibrium equation can be written as gradient of a scalar ψ [18] ~ ξ = ∇ψ . (26) Inserting this approach into Eq. (19) yields ∇ (