is given by the collisionless Boltzmann equation and the velocity dispersions of the system are calculated via the Jean's equations (e.g.Ref.[21]).Solving the Jean's equations requires assumptions to be made,and therefore even for a specific density distribution the solution is not unique.Several specific models have been used in the context of WIMP direct detection signals.The logarithmic ellipsoidal model [22]is the simplest triaxial generalisation of the isothermal sphere and has a velocity distribution which is a multi-variate Gaussian.Osipkov- Merritt models [23;24]are spherically symmetric with radially dependent anisotropic velocity distributions.Fitting functions for the speed distributions in these models are available,for a selection of density profiles,in Ref.[25].Velocity distributions have also been extracted from cosmological simulations,with both multi-variate Gaussian [26]and Tsallis [27]distri- butions [28]being advocated as fitting functions.While it is not known whether any of these models provide a good approximation to the real local velocity distribution function,the models are none the less useful for assessing the uncertainties in the direct detection signals. Particles with speed,in the Galactic rest frame,greater than the local escape speed, vesc=√2Φ(Ro)I whereΦ(r）is the potential,are not gravitationally bound.Many of the models used,in particular the standard halo model,formally extend to infinite radii and therefore their speed distribution has to be truncated at vese 'by hand'(see e.g.Ref.[29]). The standard value for the escape speed is vese=650kms-1.A recent analysis,using high velocity stars from the RAVE survey,finds 498kms-<vese 608kms-1 with a median likelihood of 544 kms-1 [30]. In Sec.4 we discuss the impact of uncertainty in the speed distribution on the direct detection signals. 3.3 Earth's motion The WIMP speed distribution in the detector rest frame is calculated by carrying out,a time dependent,Galilean transformation:v=v+ve(t).The Earth's motion relative to the Galactic rest frame,ve(t),is made up of three components:the motion of the Local Standard of Rest (LSR),the Sun's peculiar motion with respect to the LSR,ve,and the Earth's orbit about the Sun,vorb If the Milky Way is axisymmetric then the motion of the LSR is given by the local circular velocity (0,ve,0),where vc =220kms-1 is the standard value.Kerr and Lynden-Bell found, by combining a large number of independent measurements,ve=(222+20)kms-1 [17].A more recent determination,using the proper motions of Cepheids measured by Hipparcos [31], is broadly consistent:vc=(218+7)kms-1(Ro/8kpc). The Sun's peculiar motion,determined using the parallaxes and proper motions of stars in the solar neighbourhood from the Hipparcos catalogue,is ve =(10.0+0.4,5.2+0.6,7.2+ 0.4)kms-1 [32]in Galactic co-ordinates(where x points towards the Galactic center,y is the direction of Galactic rotation and z towards the North Galactic Pole). A relatively simple,and reasonably accurate,expression for the Earth's motion about the Sun can be found by ignoring the ellipticity of the Earth's orbit and the non-uniform motion of the Sun in right ascension [33]:verb =veleisin A(t)-e2 cos A(t)]where ve 29.8kms-1 is the orbital speed of the Earth,A(t)=2(t-0.218)is the Sun's ecliptic longitude(with t in years)and e1(2)are unit vectors in the direction of the Sun at the Spring equinox (Summer solstice).In Galactic co-ordinates e=(-0.0670,0.4927,-0.8676)and e2=(-0.9931,-0.1170,0.01032).More accurate expressions can be found in Ref.[34]. 7is given by the collisionless Boltzmann equation and the velocity dispersions of the system are calculated via the Jean’s equations (e.g. Ref. [21]). Solving the Jean’s equations requires assumptions to be made, and therefore even for a specific density distribution the solution is not unique. Several specific models have been used in the context of WIMP direct detection signals. The logarithmic ellipsoidal model [22] is the simplest triaxial generalisation of the isothermal sphere and has a velocity distribution which is a multi-variate Gaussian. Osipkov￾Merritt models [23; 24] are spherically symmetric with radially dependent anisotropic velocity distributions. Fitting functions for the speed distributions in these models are available, for a selection of density profiles, in Ref. [25]. Velocity distributions have also been extracted from cosmological simulations, with both multi-variate Gaussian [26] and Tsallis [27] distri￾butions [28] being advocated as fitting functions. While it is not known whether any of these models provide a good approximation to the real local velocity distribution function, the models are none the less useful for assessing the uncertainties in the direct detection signals. Particles with speed, in the Galactic rest frame, greater than the local escape speed, vesc = p 2|Φ(R0)| where Φ(r) is the potential, are not gravitationally bound. Many of the models used, in particular the standard halo model, formally extend to infinite radii and therefore their speed distribution has to be truncated at vesc ‘by hand’ (see e.g. Ref. [29]). The standard value for the escape speed is vesc = 650 km s−1 . A recent analysis, using high velocity stars from the RAVE survey, finds 498km s−1 < vesc < 608 km s−1 with a median likelihood of 544 km s−1 [30]. In Sec. 4 we discuss the impact of uncertainty in the speed distribution on the direct detection signals. 3.3 Earth’s motion The WIMP speed distribution in the detector rest frame is calculated by carrying out, a time dependent, Galilean transformation: v → v˜ = v + ve(t). The Earth’s motion relative to the Galactic rest frame, ve(t), is made up of three components: the motion of the Local Standard of Rest (LSR), the Sun’s peculiar motion with respect to the LSR, v p ⊙, and the Earth’s orbit about the Sun, v orb e . If the Milky Way is axisymmetric then the motion of the LSR is given by the local circular velocity (0, vc, 0), where vc = 220 km s−1 is the standard value. Kerr and Lynden-Bell found, by combining a large number of independent measurements, vc = (222 ± 20) km s−1 [17]. A more recent determination, using the proper motions of Cepheids measured by Hipparcos [31], is broadly consistent: vc = (218 ± 7) km s−1 (R0/8 kpc). The Sun’s peculiar motion, determined using the parallaxes and proper motions of stars in the solar neighbourhood from the Hipparcos catalogue, is v p ⊙ = (10.0 ± 0.4, 5.2 ± 0.6, 7.2 ± 0.4) km s−1 [32] in Galactic co-ordinates (where x points towards the Galactic center, y is the direction of Galactic rotation and z towards the North Galactic Pole). A relatively simple, and reasonably accurate, expression for the Earth’s motion about the Sun can be found by ignoring the ellipticity of the Earth’s orbit and the non-uniform motion of the Sun in right ascension [33]: v orb e = ve[e1 sin λ(t) − e2 cos λ(t)] where ve = 29.8 km s−1 is the orbital speed of the Earth, λ(t) = 2π(t − 0.218) is the Sun’s ecliptic longitude (with t in years) and e1(2) are unit vectors in the direction of the Sun at the Spring equinox (Summer solstice). In Galactic co-ordinates e1 = (−0.0670, 0.4927, −0.8676) and e2 = (−0.9931, −0.1170, 0.01032). More accurate expressions can be found in Ref. [34]. 7