Draft version June 1, 2021 Typeset using LATEX default style in AASTeX63 Energetic Explosions from Collisions of Stars at Relativistic Speeds in Galactic Nuclei Betty X. Hu1 and Abraham Loeb2 1Department of Physics, Harvard University 17 Oxford Street Cambridge, MA 02138, USA 2Department of Astronomy, Harvard University 60 Garden Street Cambridge, MA 02138, USA (Received June 1, 2021) Submitted to ApJ ABSTRACT We consider collisions between stars moving near the speed of light around supermassive black holes 8 (SMBHs), with mass M• & 10 M , without being tidally disrupted. The overall rate for collisions 8 9 10 −1 taking place in the inner ∼ 1 pc of galaxies with M• = 10 ; 10 ; 10 M are Γ ∼ 5; 0:07; 0:02 yr , respectively. We further calculate the differential collision rate as a function of total energy released, energy released per unit mass lost, and galactocentric radius. The most common collisions will release energies on the order of ∼ 1049 − 1051 erg, with the energy distribution peaking at higher energies in galaxies with more massive SMBHs. Depending on the host galaxy mass and the depletion timescale, the overall rate of collisions in a galaxy ranges from a small percentage to several times larger than that of core-collapse supernovae (CCSNe) for the same host galaxy. In addition, we show example light curves for collisions with varying parameters, and find that the peak luminosity could reach or even exceed that of superluminous supernovae (SLSNe), although with light curves with much shorter duration. Weaker events could initially be mistaken for low-luminosity supernovae. In addition, we note that these events will likely create streams of debris that will accrete onto the SMBH and create accretion flares that may resemble tidal disruption events (TDEs). Keywords: Supermassive black holes | stellar collisions | galaxy cores | transients 1. INTRODUCTION Supernova explosions release of order 1051 ergs of energy, originate from runaway ignition of degenerate white dwarfs (Hillebrandt & Niemeyer 2000) or the collapse of a massive star (Woosley & Weaver 1995; Barkat et al. 1967). Rubin arXiv:2105.14026v1 [astro-ph.HE] 28 May 2021 & Loeb(2011) and Balberg et al.(2013) considered a separate, rare kind of explosive event from collisions between hypervelocity stars in galactic nuclei. The cluster of stars builds up over time and reaches a steady state condition in which the rate of stellar collisions is similar to the formation rate of new stars. A simplified model for the explosion light curve with the "radiative zero" approach by Arnett (Arnett 1996), which assumes that the shocked material has uniform density and temperature and a homologous velocity profile, shows that the resulting light curve would have an average luminosity on the order of ∼ 2 × 1041 erg s−1, on par with faint conventional supernovae. Furthermore, the light curve would be expected to include a long flare due to the accretion of stellar material onto the supermassive [email protected] [email protected] 2 black hole (SMBH) at the center of the galaxy. Rubin & Loeb(2011) also considered mass loss from collisions between stars at the galactic center in order to constrain the stellar mass function. In this work, we consider high-speed stars at galactic centers. Approaching stars can be tidally disrupted by a SMBH 1=3 at the tidal-disruption radius, rT ∼ R?(M•=M?) , with R? the radius of the star and M• and M? the masses of the black hole and star, respectively. For sun-like stars, the tidal-disruption radius is smaller than the black hole's event 2 8 horizon radius rs = 2GM=c for black hole masses & 10 M (Stone et al. 2019). For maximally spinning black holes, 8 tidal disruption events (TDEs) can be observed for sun-like stars near SMBHs as large as ∼ 7×10 M (Kesden 2012). 8 In this work we consider SMBHs with masses M• & 10 M . The stars could be moving near the speed of light close to the SMBH. We adopt a Newtonian approach and ignore the effects of general relativity near the SMBH because the chances of collisions to occur in a region where they would matter are extremely small. Surveys from the last two decades such as the Sloan Digital Sky Survey (SDSS, Frieman et al. 2008), Palomar Transient Factory (PTF, Rau et al. 2009), Zwicky Transient Factory (ZTF, Bellm 2014), Pan-STARRS (Scolnic et al. 2018), and others (Guillochon et al. 2017), have greatly increased the number of supernovae detected. In addition to detecting many more already well-understood classes of supernovae, previously unheard of transients were also detected, such as superluminous supernovae (Gal-Yam 2012; Bose et al. 2018; Gal-Yam 2019), rapidly-decaying supernovae (Perets et al. 2010; Kasliwal et al. 2010; Prentice et al. 2018; Nakaoka et al. 2019; Tampo et al. 2020), and transients with slow temporal evolution (Taddia et al. 2016; Arcavi et al. 2017; Dong et al. 2020; Guti´errezet al. 2020). These discoveries have challenged existing theories of transients and suggest that a much broader range of events remain to be detected. The Vera C. Rubin observatory is expected to start operation in 2023 and to detect hundreds of thousands of supernovae a year over a ten-year survey (Ivezi´cet al. 2019). The outline of this paper is as follows. In section2, we describe how we simulate stellar collisions and calculate light curves. In section3, we provide the results of our calculations. In section4, we estimate the observed rates of our events. Finally, in section5 we summarize our main conclusions. 2. METHOD 2.1. Explosion Parameters Rubin & Loeb(2011) provide the differential collision rate between two species of stars, labeled "1" and "2", at some impact parameter b with distribution functions f1 and f2 and velocities ~v1 and ~v2, 3 3 2 dΓ = f1 (rgal;~v1) d v1f2 (rgal;~v2) d v2 × j~v1 − ~v2j2πb db 4πrgaldrgal; (1) assuming spherical symmetry, with dependence only on galocentric radius, rgal. Taking f1 and f2 as Maxwellian distributions and adopting a power-law present-day mass function (PDMF), ξ ≡ dn=dM / M −α, Eq. (1) simplifies to 2 2 3=2 −3 −vrel=4σ 3 2 −α −α 2 dΓ = 4π σ e vrelK (rgal) M1 M2 rgalb db drgal dvrel dM1 dM2: (2) The relative velocity between the stars is vrel = j~v1 −~v2j, and K (rgal) is a normalization constant which can be solved for from the density profile, M 2−α − M 2−α ρ (r ) = K (r ) max min : (3) gal gal 2 − α The stellar density profile is adapted from Tremaine et al.(1994), η rsM? ρη(rgal) ≡ ; (4) 4π 3−η 1+η rgal (rs + rgal) where we adopt the commonly-used index η = 2 (Hernquist 1990), M? is the total mass of the host spheroid, and rs is a distinctive scaling radius. We use the following relation between the mass of the black hole M• and the mass of the host spheroid M∗ (Graham 2012), M• M? log = α + β log 10 ; (5) M 7 × 10 M 8 9 10 with best-fit values α = 8:4 and β = 1:01. For M• = 10 ; 10 ; 10 M , we find spheroid masses of M? ∼ 2:8 × 10 11 12 10 ; 2:7 × 10 ; 2:7 × 10 M , respectively. Using our chosen parameters and the data from Sahu et al.(2020), we take the scaling radius as rs ∼ 0:8; 6; 50 kpc, respectively. 3 Based on Eq. (2), we define probability distribution functions (PDFs) for the parameters b, rgal, vrel, M1, and M2. We assume a Salpeter-like mass function and take α = 2:35;Mmin = 0:1 M ; and Mmax = 125 M . For the impact parameter b, we take dP=db / b, where we take bmin = 0 and bmax = R1 + R2, the sum of the radii of the colliding stars. This in turn requires the values of the two two radii R1 and R2. We use the stellar M − R relation, log R = a + b log M; (6) with a = 0:026 and b = 0:945 for M < 1:66 M , and a = 0:124 and b = 0:555 for M > 1:66 M (Demircan & Kahraman 2 1991). The PDF for the galactocentric radius rgal can be calculated from the density profile, dP=drgal / ρ (rgal) rgal, −5 where we take rmin = 10 pc and rmax = 200 pc. However, the relevant range of interest for this work (i.e. where high-velocity collisions are most likely to take place) is actually only from roughly rmin to 1 pc, the latter distance which we call rcap. We assume a Maxwellian distribution for the relative velocity vrel, dP 2 2 −1=2 −3 2 −vrel=4σ = (4π) σ vrele ; (7) dvrel where vrel can range from 0 to a fraction of the speed of light because we are not including special and general relativistic corrections; in a typical calculation with a high number of samples, the maximum relative velocity observed among them is no more than ∼ 5% the speed of light. We calculate the velocity dispersion from equations given in Tremaine et al.(1994). To run a Monte Carlo integration, we draw a fixed number N of sample values from each of the probability distri- butions. Each sample is meant to represent two stars with known masses (M1, M2) and radii (R1, R2) colliding with some know relative velocity vrel and impact parameter b at some galactocentric radius rgal. We use a Monte Carlo estimator to calculate the multidimensional integral, Z Γ = dΓ(b; rgal; vrel;M1;M2)db drgal dvrel dM1 dM2: (8) For a given collision, the kinetic energy of the ejecta is estimated from collision kinematics as, 2 2 Aint(R1;R2; b) M1M2vrel Aint(R1;R2; b) Eej = µvrel = ;: (9) Min(R1;R2) 2(M1 + M2) Min(R1;R2) with µ the reduced mass and Aint(R1;R2; b) the area of intersection of the collision.