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Ellipsars: Ring-like explosions from flattened stars

The stellar cataclysms producing astronomical transients have long been modeled as either a point-like explosion or jet-like engine ignited at the center of a spherically symmetric star. However, many stars are observed, or are expected on theoretical grounds, not to be precisely spherically symmetric, but rather to have a slightly flattened geometry similar to that of an oblate spheroid. Here we present axisymmetric two-dimensional hydrodynamical simulations of the dynamics of point-like explosions initiated at the center of an aspherical massive star with a range of oblateness. We refer to these exploding aspherical stars as"ellipsars"in reference to the elliptical shape of the iso-density contours of their progenitors in the two-dimensional axisymmetric case. We find that ellipsars are capable of accelerating expanding rings of relativistic ejecta which may lead to the production of astronomical transients including low-luminosity GRBs, relativistic supernovae, and Fast Blue Optical Transients (FBOTs.)


INTRODUCTION
Astrophysical transients are a topic of intense current interest, especially due to multiple time-domain surveys of the sky currently planned or in progress (e.g., Barthelmy et al. 2005;Shappee et al. 2014;Chambers et al. 2016;Kochanek et al. 2017;Ivezić et al. 2019;Bellm et al. 2019). These surveys are revealing an exciting variety of new transients, including low-luminosity gamma-ray bursts (llGRBs), relativistic supernovae (SNe) and Fast Blue Optical Transients (FBOTs). These phenomena have been interpreted as new types of stellar explosions requiring the ejection of a fraction of the star at relativistic or near-relativistic speed.
Jetted explosions require the extraction of rotational kinetic energy from a rotating stellar core, implying that the progenitor stars in which these explosions take place must be at least slightly aspherical. Furthermore, some massive stars have been directly observed to have en-velopes rotating near breakup (e.g., Porter & Rivinius 2003), some exhibiting oblateness parameters as large as 0.5 (Kervella & Domiciano de Souza 2006;Carciofi et al. 2008). Additionally, surveys of massive stars indicate that 70% are in close binary systems (Sana et al. 2012). These observations give rise to the possibility that many or most massive stars have tidally distorted envelopes.
It is thus reasonable on both observational and theoretical grounds to consider point-like explosions in aspherical stars, and to investigate the degree to which the explosion dynamics might be capable of producing relativistic ejecta. In this Letter, we present a suite of two-dimensional axisymmetric simulations of point explosions in an 18M pre-supernova helium star, which is flattened to varying degrees of oblateness. Due to the elliptical shape of the iso-density contours in the preexplosion star, we dub these explosions "ellipsars." Our Letter is organized as follows: Section 2 describes the numerical setup and initial conditions; in Section 3 we present our results, and Section 4 discusses the astrophysical implications and relevance of our work.

Equations of motion
Our simulations of point-like explosions of oblate stars are based on time-dependent solutions of the relativistic hydrodynamics equations, in 2D spherical-polar coordinates (r-θ) with axial symmetry (∂/∂φ = 0). In an explicitly conservative form, the equations of motion are the mass-continuity equation: ∂ µ (ρu µ ) = 0, and the conservation of energy and momentum: ∂ µ (T µν ) = 0, where T µν = ρhu µ u ν + pη µν is the stress-energy tensor for a perfect fluid.
In these equations, ρ is the proper (i.e. comoving) mass density, h = 1 + + p/ρ is the specific enthalpy, p is the gas pressure, is the specific internal energy, γ = 4/3 is the adiabatic index, η µν is the Minkowski metric with signature (−+++), and u µ is the fluid fourvelocity. The system is closed by the adiabatic equation of state p = (γ − 1)ρ .
Numerical solutions are obtained using a standard second-order Godunov method (see e.g. Martí & Müller 2003. Simulations use the generalized minmod slope limiter for piecewise linear reconstruction, with the least diffusive θ-parameter, θ = 2 and we invoke the method described in Quirk (1994) to suppress gridaligned shock instabilities in the flow. Results are presented for simulations with 5609 radial zones and 4096 angular zones, for which our results are numerically wellconverged. The equations are implemented in a new GPU-accelerated hydrodynamics code, SIMBI, which was written by this study's lead author, and is publicly available on GitHub. The code is written in a combination of Python and C++ (with CUDA or ROCm language extensions), and supports execution on multi-core CPUs (parallelized with OpenMP), and on NVIDIA and AMD GPU hardware. Our initial conditions include an oblate progenitor star with a surrounding wind-like medium, and a spherical high-pressure region at the center to initiate a point-like explosion. We use the MESA code (Paxton et al. 2011(Paxton et al. , 2013 to evolve a star before collapse, where the initial stellar model was a low metallicity 30M main sequence star, rotating near 99% of its breakup velocity. By the time the star explodes, it is an 18M Wolf-Rayet star with about half the radius of the sun. The stellar evolution model is 1D, yielding a spherically symmetric density profile for the stellar envelope. This density profile could be tabulated and accessed by the code to generate an initial condition, however the simple analytic fitting formula,

Initial Conditions
provides a close approximation to the density structure in the MESA model (see Duffell & MacFadyen 2015), and is more convenient for code implementation. Here, ρ c is central density, R 3 is the stellar radius, R 1,2 are the first and second break radii of the density profile, and k 1,2 are the power-law slopes of the density profile be-tween the breaks. Numerical values of these parameters are given in Table 1.
To generate a simple model for oblate stellar progenitors, the spherical density iso-surfaces at radius r are mapped to oblate ellipsoids. This is accomplished by replacing R 3 in Equation 1 with where ε ∈ [0, 1] is the flattening parameter. The semimajor and semi-minor axes a and b are determined from the condition that the progentitor volume ∝ a 2 b is independent of ε, and they satisfy a To model the circumstellar environment, we place the wind-like density profile outside the stellar envelope, where r > R ε (θ). The parameter A is conventionally non-dimensionalized as A = A * × 5 × 10 11 g cm −1 . In our simulations, we adopt the fiducial value A * = 0.1, motivated by values calculated for sub-solar metallicity (Z = 0.1Z ) Wolf-Rayet stars (e.g. Vink & de Koter 2005;Vink 2021). The wind speed is set to v wind = 10 3 km s −1 , which for A * = 0.1 corresponds to a mass lass rate of 10 −6 M yr −1 . We initiate the explosion by placing a high-pressure region at small radii r < r exp , where H(r) is the Heaviside step function and E exp is the explosion energy. Otherwise, the gas in the progenitor star and the wind medium is assumed to be cold, with p/ρ set to a small number around 10 −6 . The simulation domain extends in radius from r in = 0.01R to r out = 200R , and over the whole sphere in latitude, θ ∈ [0, π]. The radial zones are spaced logarithmically and such that ∆r = r∆θ. The explosion radius is r exp = 1.5r in , so the initial high-pressure region is covered by 230 radial zones. We simulate stars with flattening parameters ε equal to 0.0, 0.05, and 0.2.
3. RESULTS Figure 1 shows a 5 × 10 52 erg point explosion set off inside a star with 5% flattening, i.e. ε = 0.05. The sequence of images shows the shock evolving first inside the stellar envelope in panel (a), while it is breaking out in panel (b), and post break-out in panel (c). After the shock is first launched inside the star, it develops a prolate geometry, advancing radially faster at the Figure 1. The dynamical evolution of a 5 × 10 52 erg explosion within a 5% flattened progenitor star (shown as the elliptical dashed line with semi-major axis ∼ 4.5 × 10 10 cm.) We first see the deformed prolate shock approach the poles in panel (a). In panel (b), the lobe shocks have spread sideways towards an impending collision point in the equatorial plane. In panel (c), the lobe shocks collide and force material radially outward, forming an expanding relativistic ring. poles than at the equator. This happens because the shock acceleration is proportional to the magnitude of the density gradient (see e.g. Sakurai 1960;Matzner & McKee 1999), and the radial density distribution in the flattened star declines faster in the polar direction than in the equatorial plane.
The prolate shock first emerges from the stellar surface at the poles. After it breaks out, the pressurized post-shock region expands freely away from the breakout point, and into the dilute wind medium outside the star, forming lobes in both hemisphere. The expanding lobes form a peanut shape surrounding the star as shown in panel (b).
The lobe surfaces collide at an oblique angle at the stellar equator, producing a very high pressure behind the point where they intersect. The highly-pressurized region, which we refer to as the ring, expands fastest toward the dilute, unshocked gas ahead of the intersection point.
The highly pressurized ring, which was heated by the lobe collision, sometimes reaches ultra-relativistic speeds as it expands radially outwards. Whether it does or not depends on whether the ring can outrun the point of contact between the two lobes; when the lobe collision point moves too fast in the radial direction, the previously shocked-heated ring decelerates as it encounters high-pressure gas that was shocked more recently by the colliding lobes. However, when the lobe intersection point does not move outward too quickly, the highpressure gas can overtake the lobe intersection point, allowing the ring to accelerate freely into the dilute wind medium and become ultra-relativistic.
The radial speed of the lobe intersection point is proportional to the obliquity of the lobe shocks. If the northern and southern lobe surfaces collide head-on, the intersection point moves rapidly outwards (note that it can be superluminal), whereas, when the lobes intersect at a high obliquity, the intersection point advances not much faster than the shocks themselves. On the other hand, the head-on shock collision produces a higher pressure than the grazing collision does. This picture suggests that an optimal lobe geometry should exist, for which the shock collision is sufficiently head-on to produce extreme heating, but also sufficiently grazing that the shock-heated gas can outrun the lobe intersection point and accelerate freely in the dilute wind medium. Since the lobe geometry is determined by the flattening parameter ε, there should then be a critical value of ε which maximizes the amount of relativistic material launched in the explosion.
This expectation is consistent with the measured distribution of kinetic energy E k (> Γβ) over gas four-  . The cumulative distributions of kinetic energy, E k (> Γβ), for a suite of explosions with energies between 10 51 erg and 10 54 erg and oblateness magnitudes ε = 0, 0.05 and 0.2. These distributions were recorded at times when the gas is approximately ballistic, from top to bottom at times 93s, 35s, 22s, 12s, and 20s post explosion. For all but the largest explosion energy, stars with 5% flattening (ε = 0.05) produce faster ejecta than their spherical counterparts. In general, the spherical star ε = 0 and the over-flattened star ε = 0.2 produces less relativistic material than the optimally flattened star (ε = 0.05).
velocity Γβ, shown in Figure 2 for a range of different explosion energies and flattening parameters. A total of 15 simulations are shown in that figure: three simulations, with ε values of 0.0, 0.05, and 0.2, are run for each of five explosion energies: 10 51 , 10 52 , 5 × 10 52 , 10 53 , and 10 54 erg. More simulations were performed, but these values of the oblateness capture the main features of the explosion dynamics. We find that the stars with 5% flattening consistently accelerate more relativistic material than a spherical star, in some cases by a significant margin. For explosion energies in the range 10 51 − 10 53 erg, the cutoff of the kinetic energy distribution extends to higher speeds, by factors of several for ε = 0.05 as compared to the spherical cases. For example, in the simulations where E exp is 5 × 10 52 erg and 10 53 erg, the optimally flattened star has ∼ 10 46 erg of kinetic energy above Γβ = 10 and Γβ = 20, respectively, whereas the equivalent explosions in the spherical star had no energy in such fast material. Importantly, we also find that excessive flattening can hinder the acceleration of relativistic material. For example, as can be seen in Figure 2, simulations with ε = 0.2 produce less relativistic material than the spherical and optimally flattened cases.
The top panel of Figure 3 shows the isotropicequivalent kinetic energy, E k,iso (> Γβ) ≡ 4π × dE k (> Γβ)/dΩ, as a function of polar angle θ for representative values of Γβ, for a 5 × 10 52 erg explosion. We see that the kinetic energy in fast material, E k,iso (Γβ > 1), is concentrated around the equator, being at least a factor of 10 higher than at the equator for the Γβ > 1 material. In contrast, the total kinetic energy, E k,iso (Γβ > 0), is slightly more concentrated at high latitudes than at the equator. These differences in the latitudinal energy distribution of fast and slow material may have implications for the evolution of the remnant morphology, which we discuss in Sec. 4. The angular extent of the relativistic ring can be measured from the cutoffs of the E k,iso (Γβ > 1) shown in Figure 3; we find the ring subtends ∆θ 6 • in latitude around the equator. This corresponds to roughly 5% of the sphere.
Based on the quantity of mass launched at each latitude, and the density of the ambient wind-like medium, we can compute a nominal deceleration length scale, which is the radius at which the coasting shell sweeps up mass in the ambient medium comparable to its own, i.e., M ambient = M iso (> Γβ, θ). Note that this deceleration radius is an approximation for the bulk flow and would be smaller for the more relativistic fluid parcels by a factor 1/Γ ej . We've also ignored effects due to spreading of the ring. We find that M iso (Γβ > 1, θ = 90 • ) 6×10 27 g M ⊕ . The secondary vertical axis at the right side of the lower panel of Figure 3 shows the nominal deceleration radius corresponding to the mass of ejecta on the left vertical axis. We use the nominal deceleration length scale of the different ejecta components in Sec. 4 to speculate about the long-term evolution of the spatial distribution of the explosion ejecta in the remnant of a ring-like explosion.

Remnant morphology
The morphology of an ellipsar remnant starts out ringlike, and is expected to become more spherical over time.
The time for the remnant to lose its ring-like morphology is inversely proportional to the density of the circumburst medium. Using Eq. 5 and our result from explosion energy 5 × 10 52 erg, we estimate the deceleration time of an ejecta parcel to be where the reference values above were evaluated at the equator, and for the mildly relativistic ejecta Γβ = 0.5. The mildly relativistic mass ejected along the poles is 3 − 4 times lower; t dec,pole (Γβ > 0.5) θ=0 54 days. The deceleration time difference between the equatorial flow and the polar flow indicates that the expanding remnant should maintain its ring-like morphology for nearly a year. We note that the nominal deceleration time is a good approximation for the mildly relativistic ejecta, but for the more relativistic fluid parcels, the deceleration time scale is shorter by a factor 1/Γ ej . Since the slow ejecta containing most of the mass expand slightly faster in the polar direction than the equatorial direction, the overall late-time morphology of the ellipsar remnant is expected to be mildly prolate.

Afterglow and Polarization
Due to the ring-like morphology of the relativistic ejecta from ellipsars, the external shock that it forms as it sweeps up the surrounding medium is novel and will produce an afterglow light curve that is distinct from a jet-driven afterglow, as will be presented in a future publication. The fastest ejecta is concentrated within 6 • of the equator and beams its radiation to a great circle 4.7 4.8 The cumulative distribution of isotropicequivalent kinetic energy E k,iso (> Γβ, θ), as a function of the polar angle θ, for representative values of Γβ. The small axes at the top show E k,iso (> 0, θ), i.e. the total isotropicequivalent kinetic energy as a function of polar angle. The total energy in the domain, above each of the Γβ cutoffs 0.5, 1, 2, and 10, is 1.6 × 10 49 erg, 1.75 × 10 48 erg, 1.8 × 10 47 erg, and 3.0 × 10 45 erg, respectively. Bottom: The cumulative distribution of isotropic-equivalent mass Miso(> Γβ, θ), as a function of θ, for representative values of Γβ. The secondary vertical axis at the right shows the nominal deceleration radius r dec (Equation 5) corresponding to the mass scale at the left, when A * has the fiducial value of 0.1.
covering 5% of the sky, whereas bipolar jets with opening angle θ j = 6 • beam their radiation to only about θ 2 j /2 = 0.5% of the sky. The required rate of ellipsars to account for a given observed population of transients is thus significantly lower than the required rate for a jet-like explosion.
Ellipsar afterglows may be polarized, depending on viewing angle, perhaps analogously to the polarization effects discussed by, e.g., Gruzinov & Waxman (1999) for jet-driven afterglows. In addition, we expect the SNe light from ellipsars to be mildly linearly polarized since it originates in the photosphere of the slower stellar material which expands as a mildly prolate spheroid.

Was SN1998bw/GRB980425 an ellipsar?
The first observed connection between supernovae and GRBs was reported by Galama et al. (1998 in the error box of the low luminosity GRB 980425. GRB 980425 was the first of a set of GRBs with a γ-ray energy thousands to millions times smaller than that of classical GRBs. Subsequently, a few other GRBs with similar characteristics to GRB 980425low-redshift, small Lorentz factor, and low-luminosity -were observed which helped classify GRB 980425 as part of a distinct GRB population termed llGRBs (Liang et al. 2007;Virgili et al. 2009). Tan et al. (2001) explored the possibility that GRB980425 was produced by shock acceleration during breakout of an energetic explosion from a stripped envelope star. We compare our results with those of Tan et al. (2001) who used a less massive progenitor for their simulations such that E exp /E rest = 3.3 × 10 −3 , which is about twice the value we use for our progenitor-explosion configuration. They conclude that a spherical, compact carbon-oxygen core of mass 5M and explosion energy a few×10 52 erg is sufficient for producing the energetics of GRB 980425. Figure 2 shows that all of our 5 × 10 52 erg explosions in the spherical and flattened stars produce enough kinetic energy in Γβ > 1 ejecta to power GRB 980425 in rough agreement with Tan et al. (2001) (even though our progenitor mass of 18M is 3.6 times larger), but with an interesting caveat. Namely, point explosions in "ideally" flattened (ε 0.05) stars tend to focus their fastest ejecta into an equatorial ring with higher outflow speed than their spherical counterparts. We find that for ejecta speeds Γβ > 1, there is, on average, an order of magnitude more energy at the equator than the poles for flattened stars. We also note that our model with 5 × 10 52 erg and ε = 0.05 was capable of producing Lorentz factors Γ > 10, approaching the upper boundary for llGRBS (Liang et al. 2007). For the same explosion energy, but with ε = 0, no such highly relativistic Figure 4. The pressure and 4-velocity at t = 93 s showing a cross-section (r-θ slice) of the explosion dynamics. The expanding mildly relativistic ring, seen also in the expanded wings, is produced at the equator by the collision of expanding lobes propagating from the poles during a Eexp = 10 51 erg point explosion in an oblate star with ε = 0.05. The dashed orange ellipse shows the original size and shape of the progenitor. In the right wing, we see a steep pressure gradient point in the +r direction which is the main driver behind the equatorial flow. The left wing shows that the ring expands radially outward with mildly relativistic speed of Γβ ∼ 0.5. material was present in both our simulations as well as the simulations presented in Tan et al. (2001).

Event rates from radiating relativistic rings
We report the isotropic equivalent kinetic energy as a function of polar angle in Figure 3. Assuming the radiation of the Γβ > 10 material is strongly beamed in the forward direction, then the isotropic equivalent kinetic energy at the equator is E k,iso (> 10) θ=90 • ∼ 10 47 erg. Note that this isotropic equivalent energy is only valid for high beaming factors. At the more modest beaming factors below Γ ∼ 10, the radiation is more or less isotropic, so the effective energies for Γβ < 10 come down by a factor of 4π. For example, the true inferred energy for Γβ > 1 at the equator is roughly 6 × 10 47 erg and not 8 × 10 48 erg as Figure 3 panel (a) might suggest. Furthermore, we find that the transient is geometry-dependent like a jet, but with a viewing probability P (∆θ) = 1 4π π/2+∆θ/2 π/2−∆θ/2 dΩ ∆θ 2 .
This has the following observational implications: (1) the occurrence rates of these events are higher, but the detection rates might be comparable due to the relatively low luminosity; (2) at best, we observe the ring edge-on and receive photons from at least half of the ring until it spreads open perpendicular to our light of sight, synonymous to that of a clam shell, or at worst, we are perpendicular to said ring and receive photons some time later once the emitting ring has slowed down enough. Only at early times if viewed edge-on will this event be seen as a relativistic equatorial outflow.

Fast Blue Optical Transients
Ellipsars with SNe-like explosion energies (∼ 10 51 erg) can produce mildly relativistic explosions, similar to the speeds inferred from observations of FBOTs (see Drout et al. 2014;Margutti et al. 2019;Coppejans et al. 2020;Perley et al. 2021). These transients have shown strong polar-equatorial velocity gradients in their spectra, suggesting an aspherical explosion; and fast ejecta with v > 0.1c is inferred due to the presence of broad absorption lines. It has recently been proposed that relativistic jets powered by a long-lived engine may be responsible for producing the fast ejecta (Soker et al. 2019;Gottlieb et al. 2022;Soker 2022). Our results indicate that the ellipsar mechanism may be capable of producing similar mildly relativistic ejecta but without requiring the formation of a jet. Figure 4 shows the formation of a mildly relativistic ring with Γβ 0.5 resulting from a 10 51 erg explosion, though it carries less mass than inferred for some FBOTs.

High energy cosmic rays
If a significant fraction of massive stars are mildly oblate and explode as ellipsars in low density environments, then in the Galactic population of stellar explosions the fraction of explosion kinetic energy that is trans-relativistic is significantly increased. Ellipsars could thus be a primary source of high energy (10 15 − 10 18 eV) cosmic rays due to their production of transrelativistic ejecta (Budnik et al. 2008).

SUMMARY AND CONCLUSIONS
This work has demonstrated the following: (1) pointlike explosions in oblate stars with an optimal flattening of ε 0.05 produce more relativistic ejecta than similar explosions in spherical stars; (2) overly flattened stars with ε 0.2 diminish the production of relativistic ejecta; (3) the relativistic ejecta are focused in a ring, which expands outwards in the equatorial plane of the progenitor; (4) explosions of oblate stars might produce transients including trans-relativistic supernovae and Fast Blue Optical Transients (FBOTs), as well as low-luminosity gamma-ray bursts (llGRBs), if viewed in the equatorial plane; (5) Supernova-like ( 10 51 erg) explosions in progenitors with ε 0.2 and embedded in sufficiently dilute environments (mass loss rates 10 −5 M yr −1 for a 10 3 km s −1 wind) are capable of accelerating mildly relativistic ejecta.
MD thanks Andrei Gruzinov for many helpful discussions and acknowledges a graduate fellowship at the Kavli Institute for Theoretical Physics during which this research was completed. AM acknowledges support from NSF grant AST-1715356. The Python package CMasher, a color-vision-deficiency-friendly scientific colormap extension to matplotlib (van der Velden 2020) was used in the creation of plots.