## Main

Using the Solar abundance19 as the mean value for stars in the Galactic disk, the total mass of heavy r-process elements (A ≥ 90) in the Galaxy is Mtot, A≥90 ≈ 5 × 103M. These elements show a consistent abundance pattern in metal-poor stars18, suggesting that they are produced in a single kind of event. The total mass yields a relation between the Galactic event rate, R, and the heavy r-process mass produced in each event Mej, A≥90 (see Fig. 1):

Here 〈R〉 is the rate averaged over the age of the Galaxy and it is not necessarily the same as the present-day event rate R0. For sources related to the death of massive stars, the event rate should follow the star formation rate, which at present is lower than the average value20. For compact binary mergers, the event rate follows the star formation rate with some delay. The event rate of short gamma-ray bursts (SGRBs) (ref. 11) that probably arise from compact binary mergers6, increases with the cosmological redshift z at least up to z ≈ 0.8. In both cases, R0 may be smaller than 〈R〉 by a factor of up to approximately five.

Using the total mass alone we cannot distinguish between the high-rate/low-yield and the low-rate/high-yield sources. Measurements of abundances of short-lived radioactive r-process nuclides can, however, remove this degeneracy, as these abundances reflect the r-process production history on timescales comparable to their lifetimes (modulo the Galactic mixing timescale). Among the various radioactive nuclides, 244Pu seems most suitable: it is produced only via the r-process; the half-life of 244Pu, 81 Million years (Myr), is sufficiently short compared to the Hubble time, although long enough to allow for significant accumulation; and both current and ESS (4.6 Gyr before present; BP) 244Pu abundance in the interstellar medium (ISM) have been measured.

The inner Solar System continuously accretes interstellar dust grains21 containing recently produced live radioactive 244Pu. Wallner et al. 8 (see also Paul et al. 22) measured the accumulation of 244Pu in a deep-sea crust sample during the past 25 Myr and estimated the 244Pu flux on the Earth’s orbit as 250−205+590 cm−2 Myr−1, where the upper and lower values correspond to 2σ limits. The corresponding mean number density of 244Pu in the ISM is 6 × 10−17 cm−3 and the 2σ upper limit is 2 × 10−16 cm−3 (see Supplementary Methods). These values are significantly lower than the number density in the ISM derived from the ESS 244Pu abundance: nPu ≈ (244Pu/238U)ESSYU, ESSnISM 6 × 10−15 cm−3. The relative abundance ratio of (244Pu/238U)ESS ≈ 0.008 is estimated from fissiogenic xenon in ESS relics9 (for example, chondritic meteorites). YU, ESS = 7.3 × 10−13 is the number abundance of 238U relative to hydrogen inferred from meteorites23 (corrected from the present for 238U decay) and nISM is the mean number density of the ISM, which is typically approximately 1 cm−3.

The abundance of a radioactive nuclide at a given location around the solar circle, r, depends on the event rate density, , where ρ(r, z) is the stellar mass density in the disk, r and z are the Galactic radius and height above the Galactic plane, and M is the total stellar mass in the disk24. The abundance depends also on the mixing rate. The most relevant mixing process is turbulent diffusion (see Supplementary Methods) whose diffusion coefficient satisfies D = αvtHα kpc2/Gyr (vt/7 km s−1)(H/0.2 kpc), where vt is the typical turbulence velocity of the ISM, H is the ISM scale height, and α is the mixing length parameter. Defining τmix as the mean time between injection events at a given location which satisfies , we derive:

The median number density (the median rather than the average reflects the density that a typical observer measures) of a short-lived radioactive nuclide with a mean-life τi is:

where is the equilibrium value and Ni is the total number of the nuclide i ejected by each event. For τi τmix, a typical observer measures neq, i. For τi τmix, a typical observer measures a number density much lower than neq, i and one needs a larger yield to reach an observed value, interpreted here as the median number density. Figure 1 depicts the needed rate and yield so that the current 244Pu number density is the median value for typical values of α, vt and H (see equation (2)). This relation (blue area in Fig. 1) becomes flatter than R Mej−1 (equation (1), green band in Fig. 1) for decreasing event rates breaking the rate-yield degeneracy.

To take into account the large fluctuation in the measured number density averaged over timescales shorter than τmix, we simulate the history of the 244Pu abundance in the ISM around the solar circle over the past 7 Gyr. We take into account the radioactive decay, the turbulent diffusion process and the time evolution of the production rate. We consider here a characteristic low-rate/high-yield case of R0 = 5 Myr−1 following the SGRB rate11 evolution and a high-rate/low-yield case of R0 = 300 Myr−1 following the cosmic star formation history20 (Fig. 2). As expected the fluctuations of the low-rate/high-yield case are much larger than those of the high-rate/low-yield case. For both cases, the estimated range of number densities around 4.6 Gyr BP are consistent with the ESS values and they decrease with time following the decreasing event rate. Whereas for R0 = 5 Myr−1 the simulated values are also consistent with the current deep-sea measurements, for R0 = 300 Myr−1 the decline is insufficient even when taking the fluctuations into account. Figure 1 depicts upper and lower bounds on the event rate which consistently explain the 244Pu abundance of the ESS and the current ISM. The sources must satisfy R0 ≤ 90 Myr−1 and Mej ≥ 0.001 M. Although these limits vary slightly with different assumed parameters (see Supplementary Methods), the qualitative result that we reach is robust and independent of these choices. We conclude that unless an unidentified process suppresses the present amount of 244Pu that reaches Earth, the heavy r-process sources are dominantly low-rate/high-yield ones. The rarity of r-processing events also consistently explains the large scatter in r-elements/Fe abundance of metal-poor stars25,26,27,28.

These results are compared with astronomical observations concerning the possible sources. The low rate clearly rules out cc-SNe. The current 244Pu abundance should be larger by a factor of 5–100 to be compatible with a dominant cc-SNe source (see Supplementary Methods). Turning to compact binary mergers, Fig. 1 depicts also the merger rate estimated from known Galactic binary neutron stars10 and from the current SGRB rate11, as well as the ejected mass of r-process elements estimated from macronova candidates associated with GRB 130603B (refs 15,16) and with GRB 060614 (ref. 17). Remarkably, the rates and masses estimated here are fully consistent with those observations. In fact most of the overlap between the allowed 244Pu region and the overall r-process production range is just in this part of the astrophysical parameter phase space describing compact binary mergers and macronova ejection estimates. This result is independent of the choice of the efficiency and diffusion parameters.

Compact binary mergers, which we can conclude are the sources of heavy r-process nucleosynthesis, are also the prime candidates of sources for the gravitational-wave detectors, advanced LIGO/Virgo and KAGRA. Our estimates provide an upper limit to the expected detection rate (assuming a detection horizon distance of 200 Mpc): RGW ≤ 30 yr−1. The estimated ejected mass in each event is significant, implying that macronovae31,32,33 and radio flares34 associated with the gravitational-wave merger events will be detectable with follow-up observations.