tokens: implications for the proposed loose correlation between explosion energy and kick velocity (Bray Eldridge 2016 ; Janka 2017 ; Vigna-Gómez et al. 2018 ). Our results constitute further evidence that this correlation is not a tight one. More likely, the explosion energy only determines the maximum attainable kick velocity, below which one expects a distribution of kicks depending on the degree of unipolarity or bipolarity (Müller et al. 2019 ). This can be accounted for in a more general formulation than in Bray Eldridge ( 2016 ) by a factor that measures the explosion asymmetry (Janka 2017 ). The evolution of the spin-kick angle α in model m39 is noteworthy. Some observations have suggested that the birth spins and kicks of neutron stars tend to be aligned (Johnston et al. 2005 ; Ng Romani 2007 ; Noutsos et al. 2013 ). However, even though hydrodynamic mechanisms for spin-kick alignment have been proposed (Janka 2017 ), no trend towards spin-kick alignment has been observed so far in 3D CCSN simulations of non-rotating progenitors (Wongwathanarat et al. 2013 ; Müller et al. 2019 ). For our non-rotating explosion model y20, we also find a large angle α ≈ 70° between the spin and kick. For the rapidly rotating model m39, however, the spin-kick angle stays at about α ≈ 30°. In this case, spin-kick alignment comes about because the bipolar outflows are aligned with the initial rotation axis, which is not changed appreciably by asymmetric accretion on to the PNS. This suggests that progenitor rotation could play a role in explaining spin-kick alignment (provided that the observational evidence remains robust). However, even for model m39, the spin-kick alignment is not perfect, and is not clear whether the polar alignment of the outflows is a robust feature for rotating progenitors. In fact, some previous 3D supernova simulations of rotating progenitors rather find that the explosion develops preferably in the equatorial plane (Summa et al. 2018 ; Takiwaki Kotake 2018 ). Clearly, more simulations are required to determine the implications of rotation for the explosion morphology and spin-kick alignment. 6 GRAVITATIONAL WAVE EMISSION The gravitational-wave emission for our models, as measured by an observer at the pole and the equator, is shown in Fig. 6 . The high mass and rotation of model m39 results in high gravitational-wave amplitudes of up to 40 cm at the pole and over 100 cm at the equator. Model y20 reaches amplitudes of ∼10 cm at the pole, and ∼18 cm at the equator. Model s18np has a much lower amplitude of ∼5 cm at the equator, due to the lack of shock revival in this model, however its prompt convection signal in the direction of the equator reaches amplitudes as high as model y20. Figure 6. The gravitational-wave emission for our models as measured at the pole (left-hand panel) and the equator (right-hand panel). Top is model m39, middle is model s18np, and bottom is model y20. All models show the rise of gravitational-wave frequency
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