DETECTING THE MEMORY EFFECT FROM A MASSIVE BLACK HOLE MERGER AT THE GALACTIC CENTER THROUGH LUNAR RANGING

A gravitational wave pulse from a major merger of massive black holes at the Galactic center induces a permanent increase in the Earth-Moon separation. For black holes of mass ∼ 10 6 M ⊙ , the shift in the local gravitational potential is comparable to the Earth-Moon potential, leading to the Moon being perturbed relative to the Earth during the passage of the pulse. The permanent increase in the Earth-Moon separation is a fraction of a millimeter, measurable by lunar ranging for future merger events.


INTRODUCTION
The black hole at the Galactic Center, SgrA*, grows in part through mergers of black holes of mass M BH ∼ 10 6 M ⊙ (Micic et al. 2011;Greene et al. 2020). Here we calculate the imprint of such mergers on the Earth-Moon seperation.

RESULTS
A merger between black holes of the above mass results in a gravitational wave pulse of a characteristic duration, The mass equivalent of the radiated energy, (∆M) GW , changes the near-Earth gravitational potential by an amount, where c is the speed-of-light, d GC ≈ 8 kpc is the distance of the Galactic center from the Sun (Reid et al. 2019), and typically (∆M) GW 0.1M BH (Healy et al. 2014).
Coincidentally, this shift in gravitational potential as a result of the energy carried by the pulse happens to be comparable to the gravitational potential that binds the Moon to Earth, where M ⊕ = 6 × 10 27 g is the mass of the Earth and d Moon ≈ 4 × 10 10 cm is the Earth-Moon distance. The gravitational radiation pulse traverses the Earth-Moon system over a lightcrossing time (∆t) cross ∼ (d Moon /c) ∼ 1.3 s, during which the gravitational-potential change affects one of the objects before the other. For (∆φ) GW φ ⊕ , the temporary weakening of the gravitational binding between the Earth and the Moon during the passage period (∆t) cross leads to an increase in the Earth-Moon separation by an amount, where v Moon ≈ 1 km s −1 is the Moon's orbital speed, and the geometric calculation ignored the small eccentricity in the Moon's orbit.
The above increase in distance as a result of the motion of the Moon relative to Earth is of the same magnitude as the known "memory effect" (Zel'dovich & Polnarev 1974;Braginskii & Grishchuk 1985;Christodoulou 1991;Bieri et al. 2012), for which the permanent change in separation between free-floating objects of negligible mass initially at rest relative to each other, is also of order, (∆d Moon /d Moon ) ∼ [(∆φ) GW /c 2 ].

IMPLICATIONS
The resulting permanent change ∆d Moon ∼ 1 mm[(∆M) GW /M BH ] is above the ultimate sensitivity threshold of lunar ranging (Murphy et al. 2012), (∆d Moon /d Moon ) ∼ 10 −14 , and could be measured for future merger events. A tight binary of black holes with individual masses ∼ 2 × 10 6 M ⊙ and a separation a would merge on a timescale of ∼ 40 yr (a/10 14 cm) 4 (Peters 1964). The existence of such a binary is not ruled out by the orbits of the S-stars which are observed at much larger distances, 10 15 cm (Gualandris et al. 2010).
The permanent displacement from the memory effect would increase slightly the eccentricity of stellar binaries at wide separations 10 16 cm, but this imprint is not detectable at the precision enabled by astronomical surveys such as Gaia (Hwang et al. 2022), even when considering the increase in its amplitude with decreasing Galactocentric distance.