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tokens: ( 15 ). For θ > 0, all modes with m ≥ 1 are populated, whereas the approximation () contains only m = 2. For θ = π /2, h × vanishes identically and the relative amplitude of the modes m ≠ 2 with respect to the mode m = 2 is the largest one, reaching ∼75% for m = 3 and ∼50% for m = 4 when a = 0.98 M . Some tests of our computations, in particular comparisons with previous results by Poisson (1993a ; a = 0) and Detweiler (1978 ; a = 0.5 M and a = 0.9 M ) are presented in Appendix A . 3. Signal-to-noise ratio in the LISA detector The results in Sect. 2 are valid for any BH. We now specialize them to Sgr A* and evaluate the S/N in the LISA detector, as a function of the mass μ of the orbiting object, the orbital radius r 0 and the spin parameter a of Sgr A*. 3.1. Computation Assuming that its noise is stationary and Gaussian, a given detector is characterized by its one-sided noise power spectral density (PSD) S n ( f ). For a gravitational wave search based on the matched filtering technique, the S/N ρ is given by the following formula (see e.g. Jaranowski Królak 2012 ; Moore et al. 2015 ): where is the Fourier transform of the imprint h ( t ) of the gravitational wave on the detector, h ( t ) being a linear combination of the two polarization modes h + and h × at the detector location: In the above expression, ( t , r , θ , φ ) are the Boyer–Lindquist coordinates of the detector (“Sgr A* frame”), while F + and F × are the detector beam-pattern coefficients (or response functions), which depend on the direction (Θ, Φ) of the source with respect to the detector’s frame and on the polarization angle Ψ, the latter being the angle between the direction of constant azimuth Φ and the principal direction “+” in the wavefront plane (i.e. the axis of the h + mode or equivalently the direction of the semi-major axis of the orbit viewed as an ellipse in the detector’s sky) ( Apostolatos et al. 1994 ). For a detector like LISA, where, for high enough frequencies, the gravitational wavelength can be comparable or smaller than the arm length (2.5 Gm), the response functions F + and F × depend a priori on the gravitational wave frequency f , in addition to (Θ, Φ, Ψ) ( Robson et al. 2019 ). However for the gravitational waves considered here, a reasonable upper bound of the frequency is that of the harmonic m = 4 (say) of waves from the prograde ISCO of an extreme Kerr BH (see Fig. 4 ). From the value given by Eq. ( 5 ), this is f max = 2 × 7.9 ≃ 15.8 mHz, the multiplication by 2 taking into account the transition from m = 2 to m = 4. This value being lower than LISA’s transfer frequency f * = 19.1 mHz ( Robson et al. 2019 ), we may consider that F + and F × do not depend on f (see Fig. 2 in Robson et al. 2019 ). They are given in terms of (Θ, Φ, Ψ) by Eq. (3.12) of Cutler (1998) (with the prefactor appearing in Eq. (3.11) included in them). Generally, the function S n ( f ) considered in the LISA literature, and in particular to present the LISA sensitivity